Mosp2005 Homework[1]

2005 MOSP Homework

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Congratulations on your excellent performance on the AMC, AIME,and USAMO tests, which has earned you an invitation to attend theMath Olympiad Summer Program! This program will be an intenseand challenging opportunity for you to learn a tremendous amount ofmathematics.

To better prepare yourself for MOSP, you need to work on thefollowing homework problems, which come from last year’s NationalOlympiads, from countries all around the world. Even if some mayseem difficult, you should dedicate a significant amount of effort tothink about them—don’t give up right away. All of you are highlytalented, but you may have a disappointing start if you do not putin enough energy here. At the beginning of the program, writtensolutions will be submitted for review by MOSP graders, and you willpresent your solutions and ideas during the first few study sessions.

You are encouraged to use the email list to discuss these and otherinteresting math problems. Also, if you have any questions aboutthese homework problems, please feel free to contact us.

Zuming Feng, [email protected]

Melania Wood, ¡[email protected]¿

1Problems for the Red Group

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2 MOSP 2005 Homework Red Group

Algebra

1.1. Let a and b be nonnegative real numbers. Prove that√

2(√

a(a + b)3 + b√

a2 + b2)≤ 3(a2 + b2).

1.2. Determine if there exist four polynomials such that the sum ofany three of them has a real root while the sum of any two ofthem does not.

1.3. Let a, b, and c be real numbers. Prove that√

2(a2 + b2) +√

2(b2 + c2) +√

2(c2 + a2)

≥√

3[(a + b)2 + (b + c)2 + (c + a)2].

1.4. Let x1, x2, . . . , x5 be nonnegative real numbers such that x1 +x2 + x3 + x4 + x5 = 5. Determine the maximum value ofx1x2 + x2x3 + x3x4 + x4x5.

1.5. Let S be a finite set of positive integers such that none of themhas a prime factor greater than three. Show that the sum of thereciprocals of the elements in S is smaller than three.

1.6. Find all function f : Z→ R such that f(1) = 5/2 and that

f(x)f(y) = f(x + y) + f(x− y)

for all integers x and y.

1.7. Let x1,1, x2,1, . . . , xn,1, n ≥ 2, be a sequence of integers andassume that not all xi,1 are equal. For k ≥ 2, if sequence {xi,k}n

i=1

is defined, we define sequence {xi,k+1}ni=1 as

xi,k+1 =12

(xi,k + xi+1,k) ,

for i = 1, 2, . . . , n, (where xn+1,k = x1,k). Show that if n is oddthen there exist indices j and k such that xj,k is not an integer.

Combinatorics

1.8. Two rooks on a chessboard are said to be attacking each other ifthey are placed in the same row or column of the board.

(a) There are eight rooks on a chessboard, none of them attacksany other. Prove that there is an even number of rooks onblack fields.

MOSP 2005 Homework Red Group 3

(b) How many ways can eight mutually non-attacking rooks beplaced on the 9× 9 chessboard so that all eight rooks are onsquares of the same color.

1.9. Set S = {1, 2, . . . , 2004}. We denote by d1 the number of subsetof S such that the sum of elements of the subset has remainder7 when divided by 32. We denote by d2 the number of subset ofS such that the sum of elements of the subset has remainder 14when divided by 16. Compute d1/d2.

1.10. In a television series about incidents in a conspicuous town thereare n citizens staging in it, where n is an integer greater than 3.Each two citizens plan together a conspiracy against one of theother citizens. Prove that there exists a citizen, against whom atleast

√n other citizens are involved in the conspiracy.

1.11. Each of the players in a tennis tournament played one matchagainst each of the others. If every player won at least onematch, show that there are three players A,B, and C such thatA beats B, B beats C, and C beats A. Such a triple of player iscalled a cycle. Determine the number of maximum cycles such atournament can have.

1.12. Determine if it is possible to choose nine points in the plane suchthat there are n = 10 lines in the plane each of which passesthrough exactly three of the chosen points. What if n = 11?

1.13. Let n be a positive integer. Show that

1(n1

) +1

2(n2

) +1

3(n3

) + · · ·+ 1n(nn

)

=1

2n−1+

12 · 2n−2

+1

3 · 2n−3+ · · ·+ 1

n · 20.

1.14. A segment of length 2 is divided into n, n ≥ 2, subintervals. Asquare is then constructed on each subinterval. Assume that thesum of the areas of all such squares is greater than 1. Show thatunder this assumption one can always choose two subintervalswith total length greater than 1.

Geometry

1.15. Isosceles triangle ABC, with AB = AC, is inscribed in circle ω.Point D lies on arc B̂C not containing A. Let E be the foot of

4 MOSP 2005 Homework Red Group

perpendicular from A to line CD. Prove that BC + DC = 2DE.

1.16. Let ABC be a triangle, and let D be a point on side AB. Circleω1 passes through A and D and is tangent to line AC at A.Circle ω2 passes through B and D and is tangent to line BC atB. Circles ω1 and ω2 meet at D and E. Point F is the reflectionof C across the perpendicular bisector of AB. Prove that pointsD, E, and F are collinear.

1.17. Let M be the midpoint of side BC of triangle ABC (AB > AC),and let AL be the bisector of the angle A. The line passingthrough M perpendicular to AL intersects the side AB at thepoint D. Prove that AD + MC is equal to half the perimeter oftriangle ABC.

1.18. Let ABC be an obtuse triangle with ∠A > 90◦, and let r and R

denote its inradius and circumradius. Prove that

r

R≤ a sin A

a + b + c.

1.19. Let ABC be a triangle. Points D and E lie on sides BC and CA,respectively, such that BD = AE. Segments AD and BE meetat P . The bisector of angle BCA meet segments AD and BE atQ and R, respectively. Prove that

PQ

AD=

PR

BE.

1.20. Consider the three disjoint arcs of a circle determined by threepoints of the circle. We construct a circle around each of themidpoint of every arc which goes the end points of the arc. Provethat the three circles pass through a common point.

1.21. Let ABC be a triangle. Prove that

a2

bc+

b2

ca+

c2

ab≥ 4

(sin2 A

2+ sin2 B

2+ sin2 C

2

).

Number Theory

1.22. Find all triples (x, y, z) in integers such that x2 +y2 +z2 = 22004.

1.23. Suppose that n is s positive integer. Determine all the possiblevalues of the first digit after the decimal point in the decimalexpression of the number

√n3 + 2n2 + n.

MOSP 2005 Homework Red Group 5

1.24. Suppose that p and q are distinct primes and S is a subset of{1, 2, . . . , p−1}. Let N(S) denote the number of ordered q-tuples(x1, x2, . . . , xq) with xi ∈ S, 1 ≤ i ≤ q, such that

x1 + x2 + · · ·+ xq ≡ 0 (mod p).

1.25. Let p be an odd prime. Prove thatp−1∑

k=1

k2p−1 ≡ p(p + 1)2

(mod p2).

1.26. Find all ordered triple (a, b, c) of positive integers such that thevalue of the expression

(b− 1

a

)(c− 1

b

)(a− 1

c

)

is an integer.

1.27. Let a1 = 0, a2 = 1, and an+2 = an+1 + an for all positiveintegers n. Show that there exists an increasing infinite arithmeticprogression of integers, which has no number in common in thesequence {an}n≥0.

1.28. Let a, b, and c be pairwise distinct positive integers, which areside lengths of a triangle. There is a line which cuts both thearea and the perimeter of the triangle into two equal parts. Thisline cuts the longest side of the triangle into two parts with ratio2 : 1. Determine a, b, and c for which the product abc is minimal.

2Problems for the Blue Group

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8 MOSP 2005 Homework Blue Group

Algebra

2.1. Let a0, a1, . . . an be integers, not all zero, and all at least −1.Given a0+2a1+22a2+· · ·+2nan = 0, prove that a0+a1+· · ·+an >

0.

2.2. The sequence of real numbers {an}, n ∈ N satisfies the followingcondition: an+1 = an(an + 2) for any n ∈ N. Find all possiblevalues for a2004.

2.3. Determine all polynomials P (x) with real coefficients such that

(x3 + 3x2 + 3x + 2)P (x− 1) = (x3 − 3x2 + 3x− 2)P (x).

2.4. Find all functions f : R→ R such that

f(x3)− f(y3) = (x2 + xy + y2)(f(x)− f(y)).

2.5. Let a1, a2, . . . , a2004 be non-negative real numbers such that a1 +· · ·+ a2004 ≤ 25. Prove that among them there exist at least twonumbers ai and aj (i 6= j) such that

∣∣√ai −√aj

∣∣ ≤ 52003

.

2.6. Let c be a fixed positive integer, and {xk}∞k=1 be a sequence suchthat x1 = c and

xn = xn−1 +⌊

2xn−1 − 2n

⌋

for n ≥ 2. Determine the explicit formula of xn in terms of n andc.

(Here bxc denote the greatest integer less than or equal to x.)

2.7. Let n be a positive integer with n greater than one, and leta1, a2, . . . , an be positive integers such that a1 < a2 < · · · < an

and1a1

+1a2

+ · · ·+ 1an

≤ 1.

Prove that(

1a21 + x2

+1

a22 + x2

+ · · ·+ 1a2

n + x2

)2

≤ 12· 1a1(a1 − 1) + x2

for all real numbers x.

MOSP 2005 Homework Blue Group 9

Combinatorics

2.8. Consider all binary sequences (sequences consisting of 0’s and1’s). In such a sequence the following four types of operation areallowed: (a) 010 → 1, (b) 1 → 010, (c) 110 → 0, and (d) 0 → 110.Determine if it is possible to obtain sequence

1 00 . . . 0︸︷︷︸2003

from the sequence0 . . . 0︸︷︷︸2003

1.

2.9. Exactly one integer is written in each square of an n by n grid,n ≥ 3. The sum of all of the numbers in any 2 × 2 square iseven and the sum of all the numbers in any 3× 3 square is even.Find all n for which the sum of all the numbers in the grid isnecessarily even.

2.10. Let T be the set of all positive integer divisors of 2004100. Whatis the largest possible number of elements that a subset S of T

can have if no element of S is an integer multiple of any otherelement of S.

2.11. Consider an infinite array of integer. Assume that each integer isequal to the sum of the integers immediately above and immedi-ately to the left. Assume that there exists a row R0 such that allthe number in the row are positive. Denote by R1 the row belowrow R0, by R2 the row below row R1, and so on. Show that foreach positive integer n, row Rn cannot contain more than n zeros.

2.12. Show that for nonnegative integers m and n,(m0

)

n + 1−

(m1

)

n + 2+ · · ·+ (−1)m

(mm

)

n + m + 1

=

(n0

)

m + 1−

(n1

)

m + 2+ · · ·+ (−1)n

(nn

)

m + n + 1.

2.13. A 10 × 10 × 10 cube is made up up from 500 white unit cubesand 500 black unit cubes, arranged in such a way that every twounit cubes that shares a face are in different colors. A line is a1 × 1 × 10 portion of the cube that is parallel to one of cube’sedges. From the initial cube have been removed 100 unit cubessuch that 300 lines of the cube has exactly one missing cube.

10 MOSP 2005 Homework Blue Group

Determine if it is possible that the number of removed black unitcubes is divisible by 4.

2.14. Let S be a set of points in the plane satisfying the followingconditions:

(a) there are seven points in S that form a convex heptagon; and

(b) for any five points in S, if they form a convex pentagon, thenthere is point in S lies in the interior of the pentagon.

Determine the minimum value of the number of elements in S.

Geometry

2.15. Let ABCDEF be a equilateral convex hexagon with ∠A+∠C +∠E = ∠B + ∠D + ∠F . Prove that lines AD, BE, and CF areconcurrent.

2.16. Let ABCD be a convex quadrilateral. Let P,Q be points on sidesBC and DC respectively such that ∠BAP = ∠DAQ. Show thatthe area of triangles ABP and ADQ is equal if and only if theline through the orthocenters of these triangles is orthogonal toAC.

2.17. Circles S1 and S2 meet at points A and B. A line through A isparallel to the line through the centers of S1 and S2 and meets S1

and S2 again C and D respectively. Circle S3 having CD as itsdiameter meets S1 and S2 again at P and Q respectively. Provethat lines CP , DQ, and AB are concurent.

2.18. The incircle O of an isosoceles triangle ABC with AB = AC

meets BC,CA, AB at K,L, M respectively. Let N be the inter-section of lines OL and KM and let Q be the intersection of linesBN and CA. Let P be the foot of the perpendicular from A toBQ. If we assume that BP = AP + 2PQ, what are the possiblevalues of AB/BC?

2.19. Let ABCD be a cyclic quadrilateral such that AB ·BC = 2 ·AD ·DC. Prove that its diagonals AC and BD satisfy the inequality8BD2 ≤ 9AC2.

2.20. A circle which is tangent to sides AB and BC of triangle ABC

is also tangent to its circumcircle at point T . If I in the incenterof triangle ABC, show that ∠ATI = ∠CTI.

MOSP 2005 Homework Blue Group 11

2.21. Points E, F, G, and H lie on sides AB,BC, CD, and DA of aconvex quadrilateral ABCD such that

AE

EB· BF

FC· CG

GD· DH

HA= 1.

Points A,B, C, and D lie on sides H1E1, E1F1, F1G1, and G1H1

of a convex quadrilateral E1F1G1H1 such that E1F1 ‖ EF ,F1G1 ‖ FG, G1H1 ‖ GH, and H1E1 ‖ HE.

Given that E1AAH1

= a, express F1CCG1

in terms of a.

Number Theory

2.22. We call a natural number 3-partite if the set of its divisors canbe partitioned into 3 subsets each with the same sum. Show thatthere exist infinitely many 3-partite numbers.

2.23. Find all real numbers x such that⌊x2 − 2x

⌋+ 2 bxc = bxc2 .

(For real number x, bxc denote the greatest integer less than orequal to x.)

2.24. Prove that the equation a3−b3 = 2004 does not have any solutionsin positive integers.

2.25. Find all prime numbers p and q such that 3p4+5p4+15 = 13p2q2.

2.26. Does there exist an infinite subset S of the natural numbers suchthat for every a, b ∈ S, the number (ab)2 is divisible by a2−ab+b2?

2.27. A positive integer n is good if n can be written as the sum of 2004positive integers a1, a2, . . . , a2004 such that 1 ≤ a1 < a2 < · · · <

a2004 and ai divides ai+1 for i = 1, 2, . . . , 2003. Show that thereare only finitely many positive integers that are not good.

2.28. Let n be a natural number and f1, f2, . . . , fn be polynomials withintegers coefficients. Show that there exists a polynomial g(x)which can be factored (with at least two terms of degree at least1) over the integers such that fi(x) + g(x) cannot be factored(with at least two terms of degree at least 1) over the integers forevery i.

3Problems for the Black Group

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14 MOSP 2005 Homework Black Group

Algebra

3.1. Given real numbers x, y, z such that xyz = −1, show that

x4 + y4 + z4 + 3(x + y + z) ≥ x2

y+

x2

z+

y2

x+

y2

z+

z2

y+

z2

x.

3.2. Let x, y, z be positive real numbers and x+y + z = 1. Prove that√

xy + z +√

yz + x +√

zx + y ≥ 1 +√

xy +√

yz +√

zx.

3.3. Find all functions f : N→ N such that

(a) f(1) = 1

(b) f(n+2)+ (n2 +4n+3)f(n) = (2n+5)f(n+1) for all n ∈ N(c) f(n) divides f(m) if m > n.

3.4. Does there exist a function f : R→ R such that for all x, y ∈ R,

f(x2y + f(x + y2)) = x3 + y3 + f(xy)?

3.5. Find the smallest real number p such that the inequality√

12 + 1 +√

22 + 1 + · · ·+√

n2 + 1 ≤ 12n(n + p)

holds for all natural numbers n.

3.6. Solve the system of equations

x2 = 1y + 1

z ,

y2 = 1z + 1

x ,

z2 = 1x + 1

y .

in the real numbers.

3.7. Find all positive integers n for which there are distinct integersa1, . . . , an such that

1a1

+2a2

+ · · ·+ n

an=

a1 + · · ·+ an

2.

Combinatorics

3.8. Let X be a set with n elements and 0 ≤ k ≤ n. Let an,k

be the maximum number of permutations of the set X suchthat every two of them have at least k common components(where a common component of f and g is an x ∈ X such thatf(x) = g(x)). Let bn,k be the maximum number of permutations

MOSP 2005 Homework Black Group 15

of the set X such that every two of them have at most k commoncomponents.

(a) Show that an,k · bn,k−1 ≤ n!.

(b) Let p be prime, and find the exact value of ap,2.

3.9. A regular 2004-sided polygon is given, with all of its diagonalsdrawn. After some sides and diagonals are removed, every vertexhas at most five segments coming out of it. Prove that one cancolor the vertices with two colors such that at least 3/5 of theremaining segments have ends with different colors.

3.10. Squares of an n× n table (n ≥ 3) are painted black and white asin a chessboard. A move allows one to choose any 2 × 2 squareand change all of its squares to the opposite color. Find all suchn that there is a finite number of the moves described after whichall squares are the same color.

3.11. A convex 2004-sided polygon P is given such that no four verticesare cyclic. We call a triangle whose vertices are vertices of P thickif all other 2001 vertices of P lie inside the circumcircle of thetriangle, and thin if they all lie outside its circumcircle. Provethat the number of thick triangles is equal to the number of thintriangles.

3.12. A group consists of n tourists. Among every three of them thereare at least two that are not familiar. For any partition of thegroup into two busses, there are at least two familiar tourists inone bus. Prove that there is a tourist who is familiar with at mosttwo fifth of all the tourists.

3.13. A computer network is formed by connecting 2004 computers bycables. A set S of these computers is said to be independent if nopair of computers of S is connected by a cable. Suppose that thenumber of cables used is the minimum number possible such thatthe size of any independent set is at most 50. Let c(L) be thenumber of cables connected to computer L. Show that for anydistinct computers A and B, c(A) = c(B) if they are connectedby a cable and |c(A)−c(B)| ≤ 1 otherwise. Also, find the numberof cables used in the network.

3.14. Eight problems were given to each of 30 students. After thetest was given, point values of the problems were determined asfollows: a problem is worth n points if it is not solved by exactly


n contestants (no partial credit is given, only zero marks or fullmarks).

(a) Is it possible that the contestant having got more points thatany other contestant had also solved less problems than anyother contestant?

(b) Is it possible that the contestant having got less points thanany other contestant has solved more problems than anyother contestant?

Geometry

3.15. A circle with center O is tangent to the sides of the angle withthe vertex A at the points B and C. Let M be a point on thelarger of the two arcs BC of this circle (different from B and C)such that M does not lie on the line AO. Lines BM and CM

intersect the line AO at the points P and Q respectively. Let K

be the foot of the perpendicular drawn from P to AC and L bethe foot of the perpendicular drawn from Q to AB. Prove thatthe lines OM and KL are perpendicular.

3.16. Let I be the incenter of triangle ABC, and let A1, B1, and C1 bearbitrary points lying on segments AI,BI, and CI, respectively.The perpendicular bisectors of segments AA1, BB1, and CC1

form triangles A2B2C2. Prove that the circumcenter of triangleA2B2C2 coincides with the circumcenter of triangle ABC if andonly if I is the orthocenter of triangle A1B1C1.

3.17. Points M and M ′ are isogonal conjugates in the triangle ABC

(i.e. ∠BAM = ∠M ′AC, ∠ACM = ∠M ′CB, ∠CBM =∠M ′BA). We draw perpendiculars MP , MQ, MR and M ′P ′,M ′Q′, M ′R′ to the sides BC, AC,AB respectively. Let QR, Q′R′

and RP, R′P ′ and PQ,P ′Q′ intersect at E, F, G respectively.Show that the lines EA, FB, and GC are parallel.

3.18. Let ABCD be a convex quadrilateral and let K, L, M, N be themidpoints of sides AB, BC, CD, DA respectively. Let NL andKM meet at point T . Show that

83[DNTM ] < [ABCD] < 8[DNTM ],

where [P ] denotes area of P .

MOSP 2005 Homework Black Group 17

3.19. Let ABCD be a cyclic quadrilateral such that AB ·BC = 2 ·AD ·DC. Prove that its diagonals AC and BD satisfy the inequality8BD2 ≤ 9AC2.

3.20. Given a convex quadrilateral ABCD. The points P and Q arethe midpoints of the diagonals AC and BD respectively. The linePQ intersects the lines AB and CD at N and M respectively.Prove that the circumcircles of triangles NAP , NBQ, MQD,and MPC have a common point.

3.21. Let ABCD be a cyclic quadrilateral who interior angle at B is60 degrees. Show that if BC = CD, then CD +DA = AB. Doesthe converse hold?

Number Theory

3.22. Let n be a natural number and f1, f2, . . . , fn be polynomials withintegers coefficients. Show that there exists a polynomial g(x)which can be factored (with at least two terms of degree at least1) over the integers such that fi(x) + g(x) cannot be factored(with at least two terms of degree at least 1) over the integers forevery i.

3.23. Let a, b, c, and d be positive integers satisfy the following proper-ties:

(a) there are exactly 2004 pairs of real numbers (x, y) with 0 ≤x, y ≤ 1 such that both ax + by and cx + dy are integers

(b) gcd(a, c) = 6.

Find gcd(b, d).

3.24. For any positive integer n, the sum

1 +12

+13

+ · · ·+ 1n

is written in the lowest form pn

qn; that is pn and qn are relative

prime positive integers. Find all n such that pn is divisible by 3.

3.25. Prove that there does not exist an integer n > 1 such that n

divides 3n − 2n.

3.26. Find all integer solutions to

y2(x2 + y2 − 2xy − x− y) = (x + y)2(x− y).


3.27. Let p be a prime number, and let 0 ≤ a1 < a2 < · · · < am < p

and 0 ≤ b1 < b2 < · · · < bn < p be arbitrary integers. Denote byk the number of different remainders of ai + bj , 1 ≤ i ≤ m and1 ≤ j ≤ n, modulo p. Prove that

(i) if m + n > p, then k = p;

(ii) if m + n ≤ p, then k ≥ m + n− 1,

3.28. Let A be a finite subset of prime numbers and a be a positiveinteger. Show that the number of positive integers m for whichall prime divisors of am − 1 are in A is finite.

4Selected Problems from MOSP

2003 and 2004 Tests

20

MOSP 2005 Homework Selected Problems from MOSP 2003 and 2004 21

Algebra

4.1.

1. Let a1, a2, . . . , an, b1, b2, . . . , bn be real numbers such that

(a21+a2

2+· · ·+a2n−1)(b2

1+b22+· · ·+b2

n−1) > (a1b1+a2b2+· · ·+anbn−1)2.

Show that a21 + a2

2 + · · ·+ a2n > 1 and b2

1 + b22 + · · ·+ b2

n > 1.

4.2. Let a, b and c be positive real numbers. Prove that

a2 + 2bc

(a + 2b)2 + (a + 2c)2+

b2 + 2ca

(b + 2c)2 + (b + 2a)2+

c2 + 2ab

(c + 2a)2 + (c + 2b)2≤ 1

2.

4.3. Let a, b, and c be positive real numbers. Prove that(

a + 2b

a + 2c

)3

+(

b + 2c

b + 2a

)3

+(

c + 2a

c + 2b

)3

≥ 3.

4.4. Prove that for any nonempty finite set S, there exists a functionf : S × S → S satisfying the following conditions:

(a) for all a, b ∈ S, f(a, b) = f(b, a);

(b) for all a, b ∈ S, f(a, f(a, b)) = b;

(c) for all a, b, c, d ∈ S, f(f(a, b), f(c, d)) = f(f(a, c), f(b, d)).

4.5. Find all pairs (x, y) of real numbers with 0 < x < π2 such that

(sin x)2y

(cos x)y2/2+

(cos x)2y

(sinx)y2/2= sin(2x).

4.6. Prove that in any acute triangle ABC,

cot3 A+cot3 B+cot3 C+6 cot A cot B cot C ≥ cot A+cot B+cot C.

4.7. Let a, b, c be positive real numbers. Prove that

1a(1 + b)

+1

b(1 + c)+

1c(1 + a)

≥ 33√

abc(1 + 3

√abc

) .

4.8. Let N denote the set of positive integers. Find all functionsf : N→ N such that

f(m + n)f(m− n) = f(m2)

for m,n ∈ N.

22 MOSP 2005 Homework Selected Problems from MOSP 2003 and 2004

4.9. Let A,B, C be real numbers in the interval (0, π2 ). Prove that

sin A sin(A−B) sin(A− C)sin(B + C)

+sin B sin(B − C) sin(B −A)

sin(C + A)

+sinC sin(C −A) sin(C −B)

sin(A + B)≥ 0.

4.10. For a pair of integers a and b, with 0 < a < b < 1000, setS ⊆ {1, 2, . . . , 2003} is called a skipping set for (a, b) if for anypair of elements s1, s2 ∈ S, |s1 − s2| 6∈ {a, b}. Let f(a, b) bethe maximum size of a skipping set for (a, b). Determine themaximum and minimum values of f .

4.11. Let R denote the set of real numbers. Find all functions f : R→R such that

f(x)f(yf(x)− 1) = x2f(y)− f(x)

for all real numbers x and y.

4.12. Show that there is an infinite sequence of positive integers

a1, a2, a3, . . .

such that

(i) each positive integer occurs exactly once in the sequence, and

(ii) each positive integer occurs exactly once in the sequence|a1 − a2|, |a2 − a3|, . . . .

4.13. Let a1, a2, . . . , an, b1, b2, . . . , bn be real numbers in the interval[1, 2] with a2

1 + a22 + · · ·+ a2

n = b21 + b2

2 + · · ·+ b2n. Determine the

minimum value of constant c such that

a31

b1+

a32

b2+ · · ·+ a3

n

bn≤ c(a2

1 + a22 + · · ·+ a2

n).

4.14. Let x, y, z be nonnegative real numbers with x2 + y2 + z2 = 1.Prove that

1 ≤ z

1 + xy+

x

1 + yz+

y

1 + zx≤√

2.

4.15. Let n be a positive number, and let x1, x2, . . . , xn be positive realnumbers such that

x1 + x2 + · · ·+ xn =1x1

+1x2

+ · · ·+ 1xn

.


Prove that1

n− 1 + x1+

1n− 1 + x2

+ · · ·+ 1n− 1 + xn

≤ 1.

4.16. Let x, y, and z be real numbers. Prove that

xyz(2x+2y−z)(2y+2z−x)(2z+2x−y)+[x2+y2+z2−2(xy+yz+zx)](xy+yz+zx)2 ≥ 0.

4.17. Let R denote the set of real numbers. Find all functions f : R→R such that

f(x)f(yf(x)− 1) = x2f(y)− f(x)


Combinatorics

4.18. Let N denote the set of positive integers, and let S be a set. Thereexists a function f : N → S such that if x and y are a pair ofpositive integers with their difference being a prime number, thenf(x) 6= f(y). Determine the minimum number of elements in S.

4.19. Let n be a integer with n ≥ 2. Determine the number of non-congruent triangles with positive integer side lengths two of whichsum to n.

4.20. Jess has 3 pegs and disks of different sizes. Jess is supposed totransfer the disks from one peg to another, and the disks haveto be sorted so that for any peg the disk at the bottom is thelargest on that peg. (Discs above the bottom one may be inany order.) There are n disks sorted from largest on bottom tosmallest on top at the start. Determine the minimum number ofmoves (moving one disk at a time) needed to move the disks toanother peg sorted in the same order.

4.21. Let set S = {1, 2, . . . , n} and set T be the set of all subsets of S

(including S and the empty set). One tries to choose three (notnecessarily distinct) sets from the set T such that either two ofthe chosen sets are subsets of the third set or one of the chosenset is a subset of both of the other two sets. In how many wayscan this be done?

4.22. Let N denote the set of positive integers, and let f : N → N bea function such that f(m) > f(n) for all m > n. For positive


integer m, let g(m) denote the number of positive integers n suchthat f(n) < m. Express

m∑

i=1

g(i) +n∑

k=1

f(k)

in closed form.

4.23. Given that it is possible to use 501 discs of radius two to cover arectangular sheet of paper, determine if 2004 discs of radius onecan always cover the same piece of paper.

4.24. [7pts] The usual procedure for shuffling a deck of n cards involvesfirst splitting the deck into two blocks and then merging the twoblocks to form a single deck in such a way that the order of thecards within each block is not changed. A trivial cut in whichone block is empty is allowed.

(a) How many different permutations of a deck of n cards can beproduced by a single shuffle?

(b) How many of these single shuffle permutations can be in-verted by another such single shuffle, putting the deck backin order?

4.25. A robot is placed on an infinite square grid; it is composed ofa (finite) connected block of units occupying one square each.A valid subdivision of the robot is a partition of its units intotwo connected pieces which meet along a single unbroken linesegment. The robot moves as follows: it may divide into a validsubdivision, then one piece may slide one square sideways so thatthe result is again a valid subdivision, at which point the piecesrejoin. (See diagram for examples.)

We say a position of the robot (i.e., a connected block of squaresin the plane) is row-convex if

(a) the robot does not occupy only a single row or only a singlecolumn, and

(b) no row meets the robot in two or more separate connectedblocks.

Prove that from any row-convex position in the plane, the robotcan move to any other row-convex position in the plane.


4.26. Let n be a positive integer, and let Sn be the set of all positiveinteger divisors of n (including 1 and n). Prove that at most halfof the elements of Sn end in the digit 3.

4.27. Let S denote the set of points inside and on the boundary of aregular hexagon with side length 1 unit. Find the smallest valueof r such that the points of S can be colored in three colors insuch a way that any two points with the same color are less thanr units apart.

4.28. Let m be a positive integer. In the coordinate plane, we call apolygon A1A2 · · ·An, admissible if

(i) each side is parallel to one of the coordinate axes;

(ii) its perimeter A1A2 + A2A3 + · · ·+ AnA1 is equal to m.

(iii) for 1 ≤ i ≤ n, Ai = (xi, yi) is a lattice point;

(iv) x1 ≤ x2 ≤ · · · ≤ xn; and

(v) y1 = yn = 0, and there is k with 1 < k < m such thaty1 ≤ y2 · · · ≤ yk and yk ≥ yk+1 ≥ · · · ≥ yn.

Determine the number of admissible polygons as a function inm. (Two admissible polygons are consider distinct if one canbe obtained by the other via a composition of reflections androtations.)

4.29. In the coordinate plane, color the lattice points which have bothcoordinates even black and all other lattice points white. Let P

be a polygon with black points as vertices. Prove that any whitepoint on or inside P lies halfway between two black points, both


of which lie on or inside P .

4.30. Let n be a positive integer greater than or equal to 4. Let S

be a sphere with center O and radius 1, and let H1,H2, . . .Hn

be n hemispheres with center O and radius 1. Sphere S iscovered by hemispheres H1,H2, . . . , Hn. Prove that one canfind positive integers i1, i2, i3, i4 such that sphere S is coveredby Hi1 ,Hi2 ,Hi3 , Hi4 .

4.31. Let n be a positive integer. Consider sequences a0, a1, . . . , an suchthat ai ∈ {1, 2, . . . , n} for each i and an = a0.

(a) Call such a sequence good if for all i = 1, 2, . . . , n, ai−ai−1 6≡ i

(mod n). Suppose that n is odd. Find the number of goodsequences.

(b) Call such a sequence great if for all i = 1, 2, . . . , n, ai−ai−1 6≡i, 2i (mod n). Suppose that n is an odd prime. Find thenumber of great sequences.

4.32. For each positive integer n, let Dn be the set of all positive divisorsof 2n3n5n. Find the maximum size of a subset S of Dn in whichno element of S is a proper divisor of any other.

4.33. Let A = (0, 0, 0) be the origin in the three dimensional coordinatespace. The weight of a point is the sum of the absolute valuesof its coordinates. A point is a primitive lattice point if all itscoordinates are integers with their greatest common divisor equalto 1. A square ABCD is called a unbalanced primitive integersquare if it has integer side length and the points B and D areprimitive lattice points with different weights.

Show that there are infinitely many unbalanced primitive inte-ger squares ABiCiDi such that the plane containing the squaresare not parallel to each other.

4.34. A 2004× 2004 array of points is drawn. Find the largest integern such that it is possible to draw a convex n-sided polygon whosevertices lie on the points of the array.

Geometry

4.35. Let ABC be an acute triangle. Let A1 be the foot of theperpendicular from A to side BC, and let AB and AC be the feetof the perpendiculars from A1 to sides AB and AC, respectively.


Line `A passes through A and is perpendicular to line ABAC .Lines `B and `C are defined analogously. Prove that lines `A, `B ,and `C are concurrent.

4.36. Let n be a positive integer. Given n non-overlapping circulardiscs on a rectangular piece of paper, prove that one can cut thepiece of paper into convex polygonal pieces each of which containsexactly one disc.

4.37. Let ABC be an acute-angled scalene triangle, and let H, I, andO be its orthocenter, incenter, and circumcenter, respectively.Circle ω passes through points H, I, and O. Prove that if one ofthe vertices of triangle ABC lies on circle ω, then there is onemore vertex lies on ω.

4.38. A circle is inscribed in trapezoid ABCD, with AD ‖ BC andAD > BC. Diagonals AC and BD meet at P . Prove that∠APD is obtuse.

4.39. Let ABC be a triangle, and let M be the midpoint of side BC.The circumcircle of triangle ACM meets side AB again at D

(other than A). Points E and F lie on segments CB and CA,respectively, such that CE = EM and CF = 3FA. Suppose thatEF ⊥ BC. Prove that ∠ABC = ∠DEF .

4.40. Let ABCD be a tetrahedron such that triangles ABC, BCD,CDA, and DAB all have the same inradius. Is it necessary thatall four triangles be congruent?

4.41. A convex polygon is called balanced if for any interior point P ,the sum of distance from P to the lines containing the sides ofthe polygon is a constant. Two convex polygons A1A2 . . . An andB1B2 . . . Bn have mutually parallel sides. Prove that A1A2 . . . An

is balanced if and only if B1B2 . . . Bn is balanced.

4.42. Let ABC be a triangle with circumcircle ω, and let P be a pointinside triangle ABC. Line ` is tangent to circle ω at C. PointsD, E, F are the feet of perpendiculars from P to lines `, AC,BC,respectively. Prove that

PD ·AB = PE ·BC + PF · CA.

4.43. Let ABC be a triangle and let D be a point in its interior.Construct a circle ω1 passing through B and D and a circle ω2

passing through C and D such that the point of intersection of


ω1 and ω2 other than D lies on line AD. Denote by E and F

the points where ω1 and ω2 intersect side BC, respectively, andby X and Y the intersections of lines DF , AB and DE, AC,respectively. Prove that XY ‖ BC.

4.44. Let ABCD be a convex cyclic quadrilateral, and let bisectors of∠A and ∠B meet at point E. Points P and Q lie on sides AD andBC, respectively, such that PQ passes through E and PQ ‖ CD.Prove that AP + BQ = PQ.

4.45. Let ABC be an acute triangle with O and I as its circumcenterand incenter, respectively. The incircle of triangle ABC touchesits sides at D, E, and F respectively. Let G be the centroid oftriangle DEF . Prove that OG ≥ 7IG.

4.46. Let ABC be an acute triangle with AB 6= AC, and let D be thefoot of perpendicular from A to line BC. Point P is on altitudeAD. Rays BP and CP meet sides AC and AB at E and F ,respectively. If BFEC is cyclic, prove that P is the orthocenterof triangle ABC.

4.47. Let ABCD be a cyclic quadrilateral. Diagonals AC and BD

meet at P . Points E, F,G, and H are the feet of perpendicularsfrom P to sides AB, BC, CD, and DA, respectively. Prove thatlines BD,EH, and FG are either concurrent or parallel to eachother.

4.48. Let ABC be a triangle and let P be a point in its interior.Lines PA, PB, PC intersect sides BC, CA, AB at D, E, F ,respectively. Prove that

[PAF ] + [PBD] + [PCE] =12[ABC]

if and only if P lies on at least one of the medians of triangleABC. (Here [XY Z] denotes the area of triangle XY Z.)

4.49. In triangle ABC three distinct triangles are inscribed, similar toeach other, but not necessarily similar to triangle ABC, withcorresponding points on corresponding sides of triangle ABC.Prove that if two of these triangles share a vertex, than the thirdone does as well.

4.50. Let ABC be an isosceles right triangle with ∠A = 90◦ andAB = 1, D the midpoint of BC and E and F two other pointson the side BC. Let M be the second point of intersection of


the circumcircles of triangles ADE and ABF . Denote by N thesecond point of intersection of the line AF with the circumcircleof triangle ACE and by P the second point of intersection of theline AD with the circumcircle of triangle AMN . Find with proofthe distance from A to P .

4.51. Let ABC be a triangle. Circle ω passes through B and C andmeet sides AB and AC again at C ′ and B′, respectively. LetH and H ′ be the orthocenters of triangles ABC and AB′C ′,respectively. Prove that lines BB′, CC ′, and HH ′ are concurrent.

4.52. Let T be the set of triangles ABC for which there is a point D onBC such that segments AB, BD,AD, DC and AC have integrallengths and ∠ACD = 1

2∠ABC = 13∠ADB.

(a) Characterize all triples {a, b, c} that are sets of side lengthsof triangles in T.

(b) Find the triangle of minimum area in T.

5.53. Let ABC be a triangle. Points D,E, F are on sides BC,CA,AB,respectively, such that DC+CE = EA+AF = FB+BD. Provethat

DE + EF + FD ≥ 12(AB + BC + CA).

4.54. Let ABC be a triangle with ω and I with incircle and incenter,respectively. Circle ω touches the sides AB, BC, and CA atpoints C1, A1, and B1, respectively. Segments AA1 and BB1

meet at point G. Circle ωA is centered at A with radius AB1.Circles ωB and ωC are defined analogously. Circles ωA, ωB , andωC are externally tangent to circle ω1. Circles ωA, ωB , and ωC

are internally tangent to circle ω2. Let O1 and O2 be the centersof ω1 and ω2, respectively. Lines A1B1 and AB meet at C2, andlines A1C1 and AC meet at B2. Prove that points I,G, O1, andO2 lie on a line ` that is perpendicular to line B2C2.

4.55. Convex quadrilateral ABCD is inscribed in circle ω. The bisectorof ∠ADC of passes through the incenter of triangle ABC. LetM be an arbitrary point on arc ADC of ω. Denote by P and Q

the incenters of triangles ABM and BCM .

(a) Prove that all triangles DPQ are similar, regardless of theposition of point M .


(b) Suppose that ∠BAC = α and ∠BCA = β. Express ∠PDQ

and the ratio DP/DQ in terms of α and β.

4.56. Let ABCD be a convex quadrilateral inscribed in circle ω, whichhas center at O. Lines BA and DC meet at point P . LinePO intersects segments AC and BD at E and F , respectively.Construct point Q in a such way that QE ⊥ AC and QF ⊥ BD.Prove that triangles ACQ and BDQ have the same area.

4.57. Let AH1, BH2, and CH3 be the altitudes of an acute scalenetriangle ABC. The incircle of triangle ABC is tangent toBC, CA, and AB at T1, T2, and T3, respectively. For k = 1, 2, 3,let Pi be the point on line HiHi+1 (where H4 = H1) such thatHiTiPi is an acute isosceles triangle with HiTi = HiPi. Provethat the circumcircles of triangles T1P1T2, T2P2T3, T3P3T1 passthrough a common point.

Number Theory

4.58. Let r be an integer with 1 < r < 2003. Prove that the arithmeticprogression

{2003n + r | n = 1, 2, 3, . . .}

contains infinitely many perfect power integers.

4.59. For positive integers m and n define

φm(n) ={

φ(n) if n divides m;0 otherwise,

where φ(n) counts the number of positive integers between 1 andn which are relatively prime to n. Show that

∑d|n φm(d) =

gcd(m, n).

4.60. Let {a1, a2, . . . , aφ(2005)} be the set of positive integers less than2005 and relatively prime to 2005. Compute

∣∣∣∣∣∣

φ(2005)∏

k=1

cos( akπ

2005

)∣∣∣∣∣∣.

4.61. Find all triples of nonnegative integers (x, y, z) for which 4x +4y + 4z is the square of an integer.


4.62. Given eight distinct positive integers not exceeding 2004, provethat there are four of them, say a, b, c, and d, such that

4 + d ≤ a + b + c ≤ 4d.

4.63. Find all polynomials p(x) with integer coefficients such that foreach positive integer n, the number 2n − 1 is divisible by p(n).

4.64. Determine if there exists a polynomial Q(x) of degree at least 2with nonnegative integer coefficients such that for each prime p,Q(p) is also a prime.

4.65. Let a, b, c be nonzero integers such that both

a

b+

b

c+

c

aand

a

c+

c

b+

b

a

are integers. Prove that |a| = |b| = |c|.4.66. Let a1, a2, . . . , an be a finite integer geometrical sequence and let

k a positive integer, relatively prime to n. Prove that ak1 + ak

2 +· · ·+ ak

n is divisible by a1 + a2 + · · ·+ an.

4.67. Let m and n be positive integers such that 2m divides the numbern(n+1). Prove that 22m−2 divides the number 1k +2k + · · ·+nk,for all positive odd numbers k with k > 1.

4.68. Find all ordered triples of primes (p, q, r) such that

p | qr + 1, q | rp + 1, r | pq + 1.

4.69. Let N+0 and Q be the set of nonnegative integers and rational

numbers, respectively. Define the function f : N+0 → Q by

f(0) = 0 and

f(3n + k) = −3f(n)2

+ k, for k = 0, 1, 2.

Prove that f is one-to-one, and determine its range.

4.70. Let a1, a2, . . . , an be integers such that all the subset sums ai1 +ai2 + · · ·+ aik

, for 1 ≤ i1 < i2 < · · · < ik ≤ n, are nonzero. Provethat it is possible to partition the set of positive integers intofinitely many subsets S1, S2, . . . , Sm in a such way that if Si (1 ≤i ≤ m) has at least n elements, then a1x1 +a2x2 + · · ·+anxn 6= 0,where x1, x2, . . . , xn are arbitrary distinct elements in Si.

5Final Round of 2004 Russia

Mathematics Olympiad

33

34 MOSP 2005 Homework 2004 Russia Math Olympiad

9.1 Each lattice point in the plane is colored with one of three colors,and each of the three colors is used at least once. Show that thereexists a right-angled triangle whose vertices have three differentcolors.

9.2 Let ABCD be a quadrilateral with an inscribed circle. Theexternal angle bisectors at A and B intersect at K, the externalangle bisectors at B and C intersect at L, the external anglebisectors at C and D intersect at M , and the external anglebisectors at D and A intersect at N . Let K1, L1, M1 andN1 be the orthocenters of triangles ABK, BCL, CDM andDAN respectively. Show that quadrilateral K1L1M1N1 is aparallelogram.

9.3 There are 2004 boxes on a table, each containing a single ball.I know that some of the balls are white and that the number ofwhite balls is even. I am allowed to indicate any two boxes andask whether at least one of them contains a white ball. What isthe smallest number of questions needed to determine two boxes,each of which is guaranteed to contain a white ball?

9.4 Let n > 3 and let x1, x2, . . . , xn be positive real numbers whoseproduct is 1. Prove that

11 + x1 + x1x2

+1

1 + x2 + x2x3+ · · ·+ 1

1 + xn + xnx1> 1.

9.5 Are there pairwise distinct positive integers m, n, p, q satisfyingm + n = p + q and

√m + 3

√n =

√p + 3

√q > 2004?

9.6 There are 2004 telephones in a cabinet. Each pair of telephonesis connected by a cable, which is colored in one of four colors.Each of the four colors appears on at least one cable. Can onealways select some of the telephones so that among their pairwisecable connections exactly three different colors occur?

9.7 The natural numbers from 1 to 100 are arranged on a circlein such a way that each number is either larger than both ofits neighbors or smaller than both of its neighbors. A pair ofadjacent numbers is called “good” if when it is removed the circleof remaining numbers still has the above property. What is thesmallest possible number of good pairs?

9.8 Let ABC be an acute triangle with circumcenter O. Let T bethe circumcenter of triangle AOC and let M be the midpoint of

MOSP 2005 Homework 2004 Russia Math Olympiad 35

segment AC. Points D and E are selected on lines AB and BC

respectively such that ∠BDM = ∠BEM = ∠ABC. Show thatlines BT and DE are perpendicular.

10.1 = 9.1.

10.2 = 9.3.

10.3 Let ABCD be a quadrilateral with both an inscribed circle and acircumscribed circle. The incircle of quadrilateral ABCD touchesthe sides AB, BC, CD and DA at points K, L, M and N

respectively. The external angle bisectors at A and B intersectat K ′, the external angle bisectors at B and C intersect at L′,the external angle bisectors at C and D intersect at M ′, andthe external angle bisectors at D and A intersect at N ′. Provethat the lines KK ′, LL′, MM ′ and NN ′ pass through a commonpoint.

10.4 = 9.4.

10.5 A sequence of nonnegative rational numbers a1, a2, . . . satisfiesam + an = amn for all m,n. Show that not all elements of thesequence can be distinct.

10.6 A country has 1001 cities, each pair of which is connected bya one-way street. Exactly 500 roads begin in each city andexactly 500 roads end in each city. Now an independent republiccontaining 668 of the 1001 cities breaks off from the country.Prove that it is possible to travel between any two cities in therepublic without leaving the republic.

10.7 A triangle T is contained inside a polygon M which has a pointof symmetry. Let T ′ be the reflection of T through some pointP inside T . Prove that at least one vertex of T ′ lies in or on theboundary of M .

10.8 Does there exist a natural number n > 101000 such that 10 6 | n

and it is possible to exchange two distinct nonzero digits in thedecimal representation of n, leaving the set of prime divisors thesame?

11.1 = 10.1 = 9.1.

11.2 Let Ia and Ib be the centers of the excircles of traingle ABC

opposite A and B, respectively. Let P be a point on thecircumcircle ω of ABC. Show that the midpoint of the segmentconnecting the circumcenters of triangles IaCP and IbCP is the

36 MOSP 2005 Homework 2004 Russia Math Olympiad

center of ω.

11.3 The polynomials P (x) and Q(x) satisfy the property that for acertain polynomial R(x, y), the identity P (x)−P (y) = R(x, y)(Q(x)−Q(y)) holds. Prove that there exists a polynomial S(x) such thatP (x) = S(Q(x)).

11.4 The cells of a 9×2004 rectangular array contain the numbers 1 to2004, each 9 times. Furthermore, any two numbers in the samecolumn differ by at most 3. Find the smallest possible value forthe sum of the numbers in the first row.

11.5 Let M = {x1, . . . , x30} be a set containing 30 distinct positive realnumbers, and let An denote the sum of the products of elementsof M taken n at a time, 1 ≤ n ≤ 30. Prove that if A15 > A10,then A1 > 1.

11.6 Prove that for N > 3, there does not exist a finite set S

containing more than 2N pairwise non-collinear vectors in theplane satisfying:

(i) for any N vectors in S, there exist N − 1 more vectors in S

such that the sum of the 2N − 1 vectors is the zero vector;

(ii) for any N vectors in S, there exist N more vectors in S suchthat the sum of the 2N vectors is the zero vector.

11.7 A country contains several cities, some pairs of which are con-nected by airline flight service (in both directions). Each such pairof cities is serviced by one of k airlines, such that for each airline,all flights that the airline offers contain a common endpoint. Showthat it is possible to partition the cities into k + 2 groups suchthat no two cities from the same group are connected by a flightpath.

11.8 A parallelepiped is cut by a plane, giving a hexagon. Supposethere exists a rectangle R such that the hexagon fits in R; thatis, the rectangle R can be put in the plane of the hexagon so thatthe hexagon is completely covered by the rectangle. Show thatone of the faces of the parallelepiped also fits in R.

6Selected Problems from

Chinese IMO Team Training in2003 and 2004

38

MOSP 2005 Homework Selected Problems from Chinese IMO Team Training in 2003 and 200439

Algebra

5.1. Let x and y be positive integers with x < y. Find all possibleinteger values of

P =x3 − y

1 + xy.

5.2. Let ABC be a triangle, and let x be a nonnegative number. Provethat

ax cos A + bx cosB + cx cos C ≤ 12(ax + bx + cx).

5.3. Given integer n with n ≥ 2, determine the number of orderedn-tuples of integers (a1, a2, . . . , an) such that

(i) a1 + a2 + · · ·+ an ≥ n2; and

(ii) a21 + a2

2 + · · ·+ a2n ≤ n3 + 1.

5.4. Let a, b, and c denote the side lengths of a triangle with perimeterno greater than 2π. Prove that there is a triangle with side lengthssin a, sin b, and sin c.

5.5. Let n be an integer greater than 1. Determine the largest realnumber λ, in terms of n, such that

a2n ≥ λ(a1 + a2 + · · ·+ an−1) + 2an.

for all positive integers a1, a2, . . . , an with a1 < a2 < · · · < an.

5.6. Let {an}∞n=1 be a sequence of real numbers such that a1 = 2 andan+1 = a2

n − an + 1, for n = 1, 2, . . . . Prove that

1− 120032003

<1a1

+1a2

+ · · ·+ 1a2003

< 1.

5.7. Let a1, a2, . . . , a2n be real numbers such that∑2n−1

i=1 (ai+1−ai)2 =1. Determine the maximum value of

(an+1 + an+2 + · · ·+ a2n)− (a1 + a2 + · · ·+ an).

5.8. Let a1, a2, . . . , an be real numbers. Prove that there is a k,1 ≤ k ≤ n, such that

|k∑

i=1

ai −n∑

i=k+1

ai| ≤ max{|a1|, |a2|, . . . , |an|}.

5.9. Let n be a fixed positive integer. Determine the smallest positivereal number λ such that for any θ1, θ2, . . . , θn in the interval

40MOSP 2005 Homework Selected Problems from Chinese IMO Team Training in 2003 and 2004

(0, π2 ), if the product

tan θ1 tan θ2 · · · tan θn = 2n2 ,

then the sum

cos θ1 + cos θ2 + · · ·+ cos θn ≤ λ.

5.10. Let {ak}∞k=1 be a sequence of real numbers such that a1 = 3,a2 = 7, and a2

n+5 = an−1an+1 for n ≥ 2. Prove that if an+(−1)n

is a prime, then n = 3m for some nonnegative integer m.

5.11. Let k be a positive integer. Prove that√

k + 1 −√

k is not thereal part of the complex number z with zn = 1 for some positiveinteger n.

5.12. Let a1, a2, . . . , an be positive real numbers such that the systemof equations

− 2x1 + x2 = −a1x1,

x1 − 2x2 + x3 = −a2x2,

x2 − 2x3 + x4 = −a3x3,

. . . · · ·xn−2 − 2xn−1 + xn = −an−1xn−1,

xn−1 − 2xn = −anxn,

has a solution (x1, x2, . . . , xn) in real numbers with (x1, x2, . . . , xn) 6=(0, 0, . . . , 0). Prove that

a1 + a2 + · · ·+ an ≥ 4n + 1

.

5.13. Let a, b, c, d be positive real numbers with ab + cd = 1. Fori = 1, 2, 3, 4, points Pi = (xi, yi) are on the unit circle. Provethat

(ay1+by2+cy3+dy4)2+(ax4+bx3+cx2+dx1)2 ≤ 2(

a2 + b2

ab+

c2 + d2

cd

).

5.14. Determine all functions f, g R→ R such that

f(x + yg(x)) = g(x) + xf(y)


5.15. Let n be a positive integer, and let a0, a1, . . . , an−1 be complexnumbers with

|a0|2 + |a1|2 + · · ·+ |an−1|2 ≤ 1.


Let z1, z2, . . . zn be the (complex) roots of the polynomial

f(z) = zn + an−1zn−1 + · · ·+ a1z + a0.

Prove that|z1|2 + |z2|2 + · · ·+ |zn|2 ≤ n.

Combinatorics

5.16. Let T be a real number satisfying the property: For any non-negative real numbers a, b, c, d, e with their sum equal to 1, it ispossible to arrange them around a circle such that the products ofany two neighboring numbers are no greater than T . Determinethe minimum value of T .

5.17. Let A = {1, 2, . . . , 2002} and M = {1001, 2003, 3005}. A subsetB of A is called M -free if the sum of any pairs of elements b1 andb2 in B, b1 + b2 is not in M . An ordered pair of subset (A1, A2) iscalled a M -partition of A if A1 and A2 is a partition of A and bothA1 and A2 are M -free. Determine the number of M -partitions ofA.

5.18. Let S = (a1, a2, . . . , an) be the longest binary sequence suchthat for 1 ≤ i < j ≤ n − 4, (ai, ai+1, ai+2, ai+3, ai+4) 6=(aj , aj+1, aj+2, aj+3, aj+4). Prove that

(a1, a2, a3, a4) = (an−3, an−2, an−1, an).

5.19. For integers r, let Sr =∑n

j=1 bjzrj where bj are complex numbers

and zj are nonzero complex numbers. Prove that

|S0| ≤ n max0<|r|≤n

|Sr|.

5.20. Let n be a positive integer. Three letters (x1, x2, x3), with eachxi being an element in {R, G, B}, is called a set if x1 = x2 = x3 or{x1, x2, x3} = {R, G,B}. Claudia is playing the following game:She is given a row of n of letters a1,1, a1,2, . . . , a1,n with each lettera1,i being either A, or B, or C. Claudia then builds a triangulararray of letter with the given row as the initial row on the top.She writes down n − 1 letters a2,1, a2,2, . . . , a2,n−1 in the secondrow such that (a1,i, a1,i+1, a2,i form a set for each 1 ≤ i ≤ n− 1.She then writes down n − 2 letters in the third row by applyingthe similar rules to letter in the second row, and so on. The games


ends when she obtain the single letter an,1 in the nth row. Forexample, Claudia will obtain the following triangular array with(a1, a2, . . . , a6) = (A,C, B, C,B, A)) as the given initial row:

R G B G B R

B R R R C

G R R B

B R G

G B

R

An initial row is balanced if there are equal numbers of A,B,C

in the row. An initial row is good if a1,1, a1,n, an,1 form a set.

(a) Find all the integers n such that all possible initial sequenceare good.

(b) Find all the integers n such that all possible balanced initialsequences are good.

5.21. Let S be a set such that

(a) Each element of S is a positive integer no greater than 100;

(b) For any two distinct elements a and b in S, there exists anelement c in S such that a and c, b and c are relatively prime;and

(c) For any two distinct elements a and b in S, there exists a thirdelement d in S such that a and d, b and d are not relativelyprime to each other.

Determine the maximum number of elements S can have.

5.22. Ten people are applying for a job. The job selection committeedecides to interview the candidates one by one. The order ofcandidates being interviewed is random. Assume that all thecandidates have distinct abilities. For 1 ≤ k ≤ 10. The followingpolicies are set up within the committee:

(i) The first three candidates interviewed will not be fired;

(ii) For 4 ≤ i ≤ 9, if the ith candidate interviewed is more capablethan all the previously interviewed candidates, then thiscandidate is hired and the interview process is terminated;

(iii) The 10th candidate interviewed will be hired.


Let Pk denote the probability that the kth most able person ishired under the selection policies. Show that,

(a) P1 > P2 > · · · > P8 = P9 = P10; and

(b) that there are more than 70% chance that one of the threemost able candidates is hired and there are no more than 10%chance that one of the three least able candidates is hired.

5.23. Let A be a subset of the set {1, 2, . . . , 29} such that for any integerk and any elements a and b in A (a and b are not necessarilydistinct), a+b+30k is not the product of two consecutive integers.Find the maximum number of elements A can have.

5.24. Let set S = {(a1, a2, . . . , an) | ai ∈ R, 1 ≤ i ≤ n}, and let A bea finite subset of S. For any pair of elements a = (a1, a2, . . . , an)and b = (b1, b2, . . . , bn) in A, define d(a, b) = (|a1 − b1|, |a2 −b2|, . . . , |an−bn|) and D(A) = {d(a, b) | a, b ∈ A}. Prove that theset D(A) contains more elements than the set A does.

5.25. Let n be a positive integer, and let A1, A2, . . . An+1 be nonemptysubsets of the set {1, 2, . . . , n}. Prove that there exists nonemptyand nonintersecting index sets I1 = {i1, i2, . . . , ik} and I2 ={j1, j2, . . . , jm} such that

Ai1 ∪Ai2 ∪ · · · ∪Aik= Aj1 ∪Aj2 ∪ · · · ∪Ajm .

5.26. Integers 1, 2, . . . , 225 are arranged in a 15×15 array. In each row,the five largest numbers are colored red. In each column, the fivelargest numbers are colored blue. Prove that there are at least25 numbers are colored purple (that is, colored in both red andblue).

5.27. Let p(x) be a polynomial with real coefficients such that p(0) =p(n). Prove that there are n distinct pairs of real numbers (x, y)such that y − x is a positive integer and p(x) = p(y).

5.28. Determine if there is an n × n array with entries −1, 0, 1 suchthat the 2n sums of all the entries in each row and column areall different, where (1) n = 2003; and (2) n = 2004.

5.29. Let P be a 1000-sided regular polygon. Some of its diagonals weredrawn to obtain a triangulation the polygon P . (The region insideP is cut into triangular regions, and the diagonals drawn onlyintersect at the vertices of P .) Let n be the number of differentlengths of the drawn diagonals. Determine the minimum value of


n.

5.30. Let m and n be positive integers with m ≥ n, and let ABCD

be a rectangle with A = (0, 0), B = (m, 0), C = (m,n), andD = (0, n). Rectangle ABCD is tiled by mn unit squares. Thereis a bug in each unit square. At a certain moment, Ben shouts atthe bugs: “Move!!!” Each bug can choose independently whetheror not to follow Ben’s order and bugs do not necessarily moveto an adjacent square. After the moves are finished, Ben noticedthat if bug a and bug b were neighbors before the move, theneither they are still neighbors or they are in the same squareafter the move. (Two bugs are called neighbors if the square theyare staying share a common edge.) Prove that, after the move,there are n bugs such that the centers of the squares they arestaying are on a line of slope 1.

Geometry

5.31. Quadrilateral ABCD is inscribed in circle with AC a diameterof the circle and BD ⊥ AC. Diagonals AC and BD intersectat E. Extend segment DA through A to F . Extend segmentBA through A to G such that DG ‖ BF . Extend segmentGF through F to H such that CH ⊥ GH. Prove that pointsB,E, F,H lie on a circle.

5.32. Let ABC be a triangle. Points D, E, and F are on segments AB,AC, and DE, respectively. Prove that

3√

[BDF ] + 3√

[CEF ] ≤ 3√

[ABC]

and determine the conditions when the equality holds.

5.33. Circle ω is inscribed in convex quadrilateral ABCD, and ittouches sides AB, BC, CD, and DA at A1, B1, C1, and D1, re-spectively. Let E,F, G, and H be the midpoints of A1B1, B1C1, C1D1,and D1A1, respectively. Prove that quadrilateral EFGH is arectangle if and only if ABCD is cyclic.

5.34. Let ABCD be a cyclic convex quadrilateral with ∠A = 60◦,BC = CD = 1. Rays AB and DC meet E, and rays AD and BC

meet at F . Suppose that the the perimeters of triangles BCE

and CDF are integers. Compute the perimeter of quadrilateralABCD.


5.35. Let ABCD be a convex quadrilateral. Diagonal AC bisects∠BAD. Let E be a point on side CD. Segments BE andAC intersect at F . Extend segment DF through F to intersectsegment BC at G. Prove that ∠GAC = ∠EAC.

5.36. Four lines are given in the plane such that each three form anon-degenerate, non-equilateral triangle. Prove that, if it is truethat one line is parallel to the Euler line of the triangle formedby the other three lines, then this is true for each of the lines.

5.37. Let ABC be an acute triangle with I and H be its incenter andorthocenter, respectively. Let B1 and C1 be the midpoints of AC

and AB respectively. Ray B1I intersects AB at B2 6= B. RayC1I intersects ray AC at C2 with C2A > CA. Let K be theintersection of BC and B2C2. Prove that triangles BKB2 andCKC2 have the same area if and only if A, I, A1 are collinear,where A1 is the circumcenter of triangle BHC.

5.38. In triangle ABC, AB 6= AC. Let D be the midpoint of sideBC, and let E be a point on median AD. Let F be the footof perpendicular from E to side BC, and let P be a point onsegment EF . Let M and N be the feet of perpendiculars from P

to sides AB and AC, respectively. Prove that M, E, and N arecollinear if and only if ∠BAP = ∠PAC.

5.39. [7pts] Let ABC be a triangle with AB = AC. Let D be the footof perpendicular from C to side AB, and let M be the midpointof segment CD. Let E be the foot of perpendicular from A toline BM , and let F be the foot of perpendicular from A to lineCE. Prove that

AF ≤ AB

3.

5.40. Let ABC be an acute triangle, and let D be a point on side BC

such that ∠BAD = ∠CAD. Points E and F are the foot ofperpendiculars from D to sides AC and AB, respectively. Let H

be the intersection of segments BE and CF . The circumcircle oftriangle AFH meets line BE again at G. Prove that segmentsBG, GE,BF can be the sides of a right triangle.

5.41. Let A1A2A3A4 be a cyclic quadrilateral that also has an inscribedcircle. Let B1, B2, B3, B4, respectively, be the points on sidesA1A2, A2A3, A3A4, A4A1 at which the inscribed circle is tangent


to the quadrilateral. Prove that(

A1A2

B1B2

)2

+(

A2A3

B2B3

)2

+(

A3A4

B3B4

)2

+(

A4A1

B4B1

)2

≥ 8.

5.42. Triangle ABC is inscribed in circle ω. Line ` passes through A

and is tangent to ω. Point D lies on ray BC and point P is inthe plane such that D,A, P lie on ` in that order. Point U is onsegment CD. Line PU meets segments AB and AC at R and S,respectively. Circle ω and line PU intersect at Q and T . Provethat if QR = ST , then PQ = UT .

5.43. Two circles ω1 and ω2 (in the plane) meet at A and B. PointsP and Q are on ω1 and ω2, respectively, such that line PQ istangent to both ω1 and ω2, and B is closer to line PQ than A.Triangle APQ is inscribed in circle ω3. Point S is such that linesPS and QS are tangent to ω3 at P and Q, respectively. Point H

is the image of B reflecting across line PQ. Prove that A,H, S

are collinear.

5.44. Convex quadrilateral ABCD is inscribed in circle ω. Let P bethe intersection of diagonals AC and BD. Lines AB and CD

meet at Q. Let H be the orthocenter of triangle ADQ. Let M

and N be the midpoints of diagonals AC and BD, respectively.Prove that MN ⊥ PH.

5.45. Let ABC be a triangle with I as its incenter. Circle ω is centeredat I and lies inside triangle ABC. Point A1 lies on ω such thatIA1 ⊥ BC. Points B1 and C1 are defined analogously. Provethat lines AA1, BB1, and CC1 are concurrent.

Number Theory

5.46. Given integer a with a > 1, an integer m is good if m =200ak + 4 for some integer k. Prove that, for any integer n,there is a degree n polynomial with integer coefficients such thatp(0), p(1), . . . , p(n) are distinct good integers.

5.47. Find all the ordered triples (a,m, n) of positive integers such thata ≥ 2, m ≥ 2, and am + 1 divides an + 203.

5.48. Determine if there exists a positive integer n such that n hasexactly 2000 prime divisors, n is not divisible by a square of aprime number, and 2n + 1 is divisible by n.


5.49. [7pts] Determine if there exists a positive integer n such that n

has exactly 2000 prime divisors, n is not divisible by a square ofa prime number, and 2n + 1 is divisible by n.

5.50. A positive integer u if called boring if there are only finitelymany triples of positive integers (n, a, b) such that n! = ua − ub.Determine all the boring integers.

5.51. Find all positive integers n, n > 1, such that all the divisors ofn, not including 1, can be written in the form of ar + 1, where a

and r are positive integers with r > 1.

5.52. Determine all positive integers m satisfying the following prop-erty: There exists prime pm such that nm−m is not divisible bypm for all integers n.

5.53. Let m and n be positive integers. Find all pairs of positive integers(x, y) such that

(x + y)m = xn + yn.

5.54. Let p be a prime, and let a1, a2, . . . , ap+1 be distinct positiveintegers. Prove that there are indices i and j, 1 ≤ i < j ≤ p + 1,such that

max{ai, aj}gcd(ai, aj)

≥ p + 1.

5.55. For positive integer n = pa11 pa2

2 · pamm , where p1, p2, . . . , pn are

distinct primes and a1, a2, . . . , am are positive integers, d(n) =max{pa1

1 , pa22 , . . . , pam

m } is called the greatest prime power divisorof n. Let n1, n2, . . . , and n2004 be distinct positive integers withd(n1) = d(n2) = · · · = d(n2004). Prove that there exists integersa1, a2, . . . , and a2004 such that infinite arithmetic progressions{ai, ai +ni, ai +2ni, . . . }, i = 1, 2, . . . , 2004, are pairwise disjoint.

5.56. Let N be a positive integer such that for all integers n > N ,the set Sn = {n, n + 1, . . . , n + 9} contains at least one numberthat has at least three distinct prime divisors. Determine theminimum value of N .

5.57. Determine all the functions from the set of positive integers tothe set of real numbers such that

(a) f(n + 1) ≥ f(n) for all positive integers n; and

(b) f(mn) = f(m)f(n) for all relatively prime positive integersm and n.


5.58. Let n be a positive integer. Prove that there is a positive integerk such that digit 7 appears for at least b2n/3c times among thelast n digits of the decimal representation of 2k.

5.59. An n×n matrix whose entries come from the set S = {1, 2, . . . , 2n−1} is called a silver matrix if, for each i = 1, 2, . . . , n, the ith rowand ith column together contain all elements of S. Determine allthe values n for which silver matrices exist.

5.60. Let n be a positive integer. Determine the largest integer f(n)such that (

2n+1

2n

)−

(2n

2n−1

)

is divisible by 2f(n).

7Selected Problems from

Romania Contests in 2004

50

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Mosp2005 Homework[1]