54th Polish Mathematical Olympiad Problems 2003

54th Polish Mathematical Olympiad Problems 2003

A1.  ABC is acute-angled. M is the midpoint of AB. A line through M meets the lines CA, CB at K, L with CK = CL. O is the circumcenter of CKL and CD is an altitude of ABC. Show that OD = OM.

A2.  0 ≤ k₁ < k₂ < ... < k_n are integers. 0 < a < 1 is a real. Show that (1-a)(a^k₁ + a^k₂ + ... + a^k_n)² < (1+a)(a^2k₁ + a^2k₂ + ... + a^2k_n).

A3.  Find all polynomials p(x) with integer coefficients such that p(n) divides 2ⁿ - 1 for n = 1, 2, 3, ... .

B1.  p is a prime and a, b, c, are distinct positive integers less than p such that a³ = b³ = c³ mod p. Show that a² + b² + c² is divisible by a + b + c.

B2.  ABCD is a tetrahedron. The insphere touches the face ABC at H. The exsphere opposite D (which also touches the face ABC and the three planes containing the other faces) touches the face ABC at O. If O is the circumcenter of ABC, show that H is the orthocenter of ABC.

B3.  n is even. Show that there is a permutation a₁a₂...a_n of 12...n such that a_i+1 ∈ {2a_i, 2a_i-1, 2a_i-n, 2a_i-n-1} for i = 1, 2, ... , n (and we use the cyclic subscript convention, so that a_n+1 means a₁).

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53rd Polish Mathematical Olympiad Problems 2002

53rd Polish Mathematical Olympiad Problems 2002

A1.  Find all triples of positive integers (a, b, c) such that a² + 1 and b² + 1 are prime and (a² + 1)(b² + 1) = c² + 1.

A2.  ABC is an acute-angled triangle. BCKL, ACPQ are rectangles on the outside of two of the sides and have equal area. Show that the midpoint of PK lies on the line through C and the circumcenter.

A3.  Three non-negative integers are written on a blackboard. A move is to replace two of the integers by their sum and (non-negative) difference. Can we always get two zeros by a sequence of moves?

B1.  Given any finite sequence x1, x2, ... , xn of at least 3 positive integers, show that either ∑₁ⁿ x_i/(x_i+1 + x_i+2) ≥ n/2 or ∑ ₁ⁿ x_i/(x_i-1 + x_i-2) ≥ n/2. (We use the cyclic subscript convention, so that x_n+1 means x₁ and x_-1 means x_n-1 etc).

B2.  ABC is a triangle. A sphere does not intersect the plane of ABC. There are 4 points K, L, M, P on the sphere such that AK, BL, CM are tangent to the sphere and AK/AP = BL/BP = CM/CP. Show that the sphere touches the circumsphere of ABCP.

B3.  k is a positive integer. The sequence a₁, a₂, a₃, ... is defined by a₁ = k+1, a_n+1 = a_n² - ka_n + k. Show that a_m and a_n are coprime (for m ≠ n).

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52nd Polish Mathematical Olympiad Problems 2001

52nd Polish Mathematical Olympiad Problems 2001

A1.  Show that x₁ + 2x₂ + 3x₃ + ... + nx_n ≤ ½n(n-1) + x₁ + x₂² + x₃³ + ... + x_nⁿ for all non-negative reals x_i.

A2.  P is a point inside a regular tetrahedron with edge 1. Show that the sum of the distances from P to the vertices is at most 3.

A3.  The sequence x₁, x₂, x₃, ... is defined by x₁ = a, x₂ = b, x_n+2 = x_n+1 + x_n, where a and b are reals. A number c is a repeated value if it occurs in the sequence more than once. Show that we can choose a, b so that the sequence has more than 2000 repeated values, but not so that it has infinitely many repeated values.

B1.  a and b are integers such that 2ⁿa + b is a square for all non-negative integers n. Show that a = 0.

B2.  ABCD is a parallelogram. K is a point on the side BC and L is a point on the side CD such that BK·AD = DL·AB. DK and BL meet at P. Show that ∠DAP = ∠BAC.

B3.  Given a set of 2000 distinct positive integers under 10¹⁰⁰, show that one can find two non-empty disjoint subsets which have the same number of elements, the same sum and the same sum of squares.

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51st Polish Mathematical Olympiad Problems 2000

51st Polish Mathematical Olympiad Problems 2000

A1.  How many solutions in non-negative reals are there to the equations:
x₁ + x_n² = 4x_n
x₂ + x₁² = 4x₁
...
x_n + x_n-1² = 4x_n-1?

A2.  The triangle ABC has AC = BC. P is a point inside the triangle such that ∠PAB = ∠PBC. M is the midpoint of AB. Show that ∠APM + ∠BPC = 180^o.

A3.  The sequence a₁, a₂, a₃, ... is defined as follows. a₁ and a₂ are primes. a_n is the greatest prime divisor of a_n-1 + a_n-2 + 2000. Show that the sequence is bounded.

B1.  PA₁A₂...A_n is a pyramid. The base A₁A₂...A_n is a regular n-gon. The apex P is placed so that the lines PA_i all make an angle 60^o with the plane of the base. For which n is it possible to find B_i on PA_i for i = 2, 3, ... , n such that A₁B₂ + B₂B₃ + B₃B₄ + ... + B_n-1B_n + B_nA₁ < 2A₁P?

B2.  For each n ≥ 2 find the smallest k such that given any subset S of k squares on an n x n chessboard we can find a subset T of S such that every row and column of the board has an even number of squares in T.

B3.  p(x) is a polynomial of odd degree which satisfies p(x²-1) = p(x)² - 1 for all x. Show that p(x) = x.

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50th Polish Mathematical Olympiad Problems 1999

50th Polish Mathematical Olympiad Problems 1999

A1.  D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the side AC such that AE/EC = BD/(AD-BC). Show that AD > BE.

A2.  Given 101 distinct non-negative integers less than 5050 show that one choose four a, b, c, d such that a + b - c - d is a multiple of 5050.

A3.  Show that one can find 50 distinct positive integers such that the sum of each number and its digits is the same.

B1.  For which n do the equations have a solution in integers:

x₁² + x₂² + 50 = 16x₁ + 12x₂

x₂² + x₃² + 50 = 16x₂ + 12x₃

...

x_n-1² + x_n² + 50 = 16x_n-1 + 12x_n

x_n² + x₁² + 50 = 16x_n + 12x₁

B2.  Show that ∑_1≤i<j≤n (|a_i-a_j| + |b_i-b_j|) ≤ ∑_1≤i<j≤n |a_i-b_j| for all integers a_i, b_i.

B3.  The convex hexagon ABCDEF satisfies ∠A + ∠C + ∠E = 360^o and AB·CD·EF = BC·DE·FA. Show that AB·FD·EC = BF·DE·CA.

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49th Polish Mathematical Olympiad Problems 1998

49th Polish Mathematical Olympiad Problems 1998

A1.  Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.

A2.  Fn is the Fibonacci sequence F₀ = F₁ = 1, F_n+2 = F_n+1 + F_n. Find all pairs m > k ≥ 0 such that the sequence x₀, x₁, x₂, ... defined by x₀ = F_k/F_m and x_n+1 = (2x_n - 1)/(1 - x_n) for x_n ≠ 1, or 1 if x_n = 1, contains the number 1.

A3.  PABCDE is a pyramid with ABCDE a convex pentagon. A plane meets the edges PA, PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P. For each of the quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the diagonals. Show that the five intersections are coplanar.

B1.  Define the sequence a₁, a₂, a₃, ... by a₁ = 1, a_n = a_n-1 + a_[n/2]. Does the sequence contain infinitely many multiples of 7?

B2.  The points D, E on the side AB of the triangle ABC are such that (AD/DB)(AE/EB) = (AC/CB)². Show that ∠ACD = ∠BCE.

B3.  S is a board containing all unit squares in the xy plane whose vertices have integer coordinates and which lie entirely inside the circle x² + y² = 1998². +1 is written in each square of S. An allowed move is to change the sign of every square in S in a given row, column or diagonal. Can we end up with all -1s by a sequence of allowed moves?

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48th Polish Mathematical Olympiad Problems 1997

48th Polish Mathematical Olympiad Problems 1997

A1.  The positive integers x₁, x₂, ... , x₇ satisfy x₆ = 144, x_n+3 = x_n+2(x_n+1+x_n) for n = 1, 2, 3, 4. Find x₇.

A2.  Find all real solutions to 3(x² + y² + z²) = 1, x²y² + y²z² + z2x² = xyz(x + y + z)³.

A3.  ABCD is a tetrahedron. DE, DF, DG are medians of triangles DBC, DCA, DAB. The angles between DE and BC, between DF and CA, and between DG and AB are equal. Show that area DBC ≤ area DCA + area DAB.

B1.  The sequence a₁, a₂, a₃, ... is defined by a₁ = 0, a_n = a_[n/2] + (-1)^n(n+1)/2. Show that for any positive integer k we can find n in the range 2^k ≤ n < 2^k+1 such that a_n = 0.

B2.  ABCDE is a convex pentagon such that DC = DE and ∠C = ∠E = 90^o. F is a point on the side AB such that AF/BF = AE/BC. Show that ∠FCE = ∠FDE and ∠FEC = ∠BDC.

B3.  Given any n points on a unit circle show that at most n²/3 of the segments joining two points have length > √2.

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47th Polish Mathematical Olympiad Problems 1996

47th Polish Mathematical Olympiad Problems 1996

A1.  Find all pairs (n,r) with n a positive integer and r a real such that 2x²+2x+1 divides (x+1)ⁿ - r.

A2.  P is a point inside the triangle ABC such that ∠PBC = ∠PCA < ∠PAB. The line PB meets the circumcircle of ABC again at E. The line CE meets the circumcircle of APE again at F. Show that area APEF/area ABP does not depend on P.

A3.  a_i, x_i are positive reals such that a₁ + a₂ + ... + a_n = x₁ + x₂ + ... + x_n = 1. Show that 2 ∑_i<j x_ix_j ≤ (n-2)/(n-1) + ∑ a_ix_i²/(1-a_i). When do we have equality?

B1.  ABCD is a tetrahedron with ∠BAC = ∠ACD and ∠ABD = ∠BDC. Show that AB = CD.

B2.  Let p(k) be the smallest prime not dividing k. Put q(k) = 1 if p(k) = 2, or the product of all primes < p(k) if p(k) > 2. Define the sequence x₀, x₁, x₂, ... by x₀ = 1, x_n+1 = xnp(x_n)/q(x_n). Find all n such that x_n = 111111.

B3.  Let S be the set of permutations a₁a₂...a_n of 123...n such that a_i ≥ i. An element of S is chosen at random. Find all n such that the probability that the chosen permutation satisfies a_i ≤ i+1 exceeds 1/3.

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46th Polish Mathematical Olympiad Problems 1995

46th Polish Mathematical Olympiad Problems 1995

 
A1.  How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1?
A2.  The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?

A3.  p ≥ 5 is prime. The sequence a₀, a₁, a₂, ... is defined by a₀ = 1, a₁ = 1, ... , a_p-1 = p-1 and a_n = a_n-1 + a_n-p for n ≥ p. Find a_p³ mod p.

B1.  The positive reals x₁, x₂, ... , x_n have harmonic mean 1. Find the smallest possible value of x₁ + x₂²/2 + x₃³/3 + ... + x_nⁿ/n.

B2.  An urn contains n balls labeled 1, 2, ... , n. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by k. Find all k such that the expected number of balls removed is k.

B3.  PA, PB, PC are three rays in space. Show that there is just one pair of points B', C' with B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA + AC'.

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45th Polish Mathematical Olympiad Problems 1994

45th Polish Mathematical Olympiad Problems 1994

A1.  Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are all integers.

A2.  L, L' are parallel lines. C is a circle that does not intersect L. A is a variable point on L. The two tangents to C from A meet L' in two points with midpoint M. Show that the line AM passes through a fixed point (as A varies).

A3.  k is a fixed positive integer. Let a_n be the number of maps f from the subsets of {1, 2, ... , n} to {1, 2, ... , k} such that for all subsets A, B of {1, 2, ... , n} we have f(A ∩ B) = min(f(A), f(B)). Find lim_n→∞ a_n^1/n.

B1.  m, n are relatively prime. We have three jugs which contain m, n and m+n liters. Initially the largest jug is full of water. Show that for any k in {1, 2, ... , m+n} we can get exactly k liters into one of the jugs.

B2.  A parallelepiped has vertices A₁, A₂, ... , A₈ and center O. Show that 4 ∑ |OA_i|² ≤ (∑|OA_i|)².

B3.  The distinct reals x₁, x₂, ... , x_n (n > 3) satisfy ∑ x_i = 0, &sum x_i² = 1. Show that four of the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ x_i³ ≤ a + b + d + nabd.

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44th Polish Mathematical Olympiad Problems 1993

44th Polish Mathematical Olympiad Problems 1993

A1.  Find all rational solutions to:
t² - w² + z² = 2xy
t² - y² + w² = 2xz
t² - w² + x² = 2yz.

A2.  A circle center O is inscribed in the quadrilateral ABCD. AB is parallel to and longer than CD and has midpoint M. The line OM meets CD at F. CD touches the circle at E. Show that DE = CF iff AB = 2CD.

A3.  g(k) is the greatest odd divisor of k. Put f(k) = k/2 + k/g(k) for k even, and 2^(k+1)/2 for k odd. Define the sequence x₁, x₂, x₃, ... by x₁ = 1, x_n+1 = f(x_n). Find n such that x_n = 800.

B1.  P is a convex polyhedron with all faces triangular. The vertices of P are each colored with one of three colors. Show that the number of faces with three vertices of different colors is even.

B2.  Find all real-valued functions f on the reals such that f(-x) = -f(x), f(x+1) = f(x) + 1 for all x, and f(1/x) = f(x)/x² for x ≠ 0.

B3.  Is the volume of a tetrahedron determined by the areas of its faces and its circumradius?

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43rd Polish Mathematical Olympiad Problems 1992

43rd Polish Mathematical Olympiad Problems 1992

A1.  Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the triangle PAB. Show that OP and CD are perpendicular.

A2.  Find all functions f : Q⁺ → Q⁺, where Q⁺ is the positive rationals, such that f(x+1) = f(x) + 1 and f(x³) = f(x)³ for all x.

A3.  Show that for real numbers x₁, x₂, ... , x_n we have ∑_i=1^m (∑_j=1ⁿ x_ix_j/(i+j) ) ≥ 0. When do we have equality?

B1.  The functions f₀, f₁, f₂, ... are defined on the reals by f₀(x) = 8 for all x, f_n+1(x) = √(x² + 6f_n(x)). For all n solve the equation f_n(x) = 2x.

B2.  The base of a regular pyramid is a regular 2n-gon A₁A₂...A_2n. A sphere passes through the apex S of the pyramid and cuts the edge SA_i at B_i (for i = 1, 2, ... , 2n). Show that ∑ SB_2i-1 = ∑ SB_2i.

B3.  Show that k³! is divisible by (k!)^k2+k+1.

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42nd Polish Mathematical Olympiad Problems 1991

42nd Polish Mathematical Olympiad Problems 1991

A1.  Do there exist tetrahedra T₁, T₂ such that (1) vol T₁ > vol T₂, and (2) every face of T₂ has larger area than any face of T₁?


A2.  Let F(n) be the number of paths P₀, P₁, ... , P_n of length n that go from P₀ = (0,0) to a lattice point P_n on the line y = 0, such that each P_i is a lattice point and for each i < n, P_i and P_i+1 are adjacent lattice points a distance 1 apart. Show that F(n) = (2n)Cn.

A3.  N is a number of the form ∑_k=1⁶⁰ a_k k^kk, where each a_k = 1 or -1. Show that N cannot be a 5th power.

B1.  Let V be the set of all vectors (x,y) with integral coordinates. Find all real-valued functions f on V such that (a) f(v) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all perpendicular v, w ∈ V. (The vector (0,0) is considered to be perpendicular to any vector.)

B2.  k₁, k₂ are circles with different radii and centers K₁, K₂. Neither lies inside the other, and they do not touch or intersect. One pair of common tangents meet at A on K₁K₂, the other pair meet at B on K₁K₂. P is any point on k₁. Show that there is a diameter of K₂ with one endpoint on the line PA and the other on the line PB.

B3.  The real numbers x, y, z satisfy x² + y² + z² = 2. Show that x + y + z ≤ 2 + xyz. When do we have equality?

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41st Polish Mathematical Olympiad Problems 1990

41st Polish Mathematical Olympiad Problems 1990

A1.  Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x²-y²) for all x, y.

A2.  For n > 1 and positive reals x1, x2, ... , xn, show that x₁²/(x₁²+x²x₃) + x₂²/(x₂²+x³x₄) + ... + x_n²/(x_n²+x¹x₂) ≤ n-1.

A3.  In a tournament there are n players. Each pair of players play each other just once. There are no draws. Show that either (1) one can divide the players into two groups A and B, such that every player in A beat every player in B, or (2) we can label the players P₁, P₂, ... , P_n such that P_i beat P_i+1 for i = 1, 2, ... n (where we use cyclic subscripts, so that P_n+1 means P₁).

B1.  A triangle with each side length at least 1 lies inside a square side 1. Show that the center of the square lies inside the triangle.

B2.  a₁, a₂, a₃, ... is a sequence of positive integers such that lim_n→∞ n/a_n = 0. Show that we can find k such that there are at least 1990 squares between a₁ + a₂ + ... + a_k and a₁ + a₂ + ... + a_k+1.

B3.  Show that ∑_k=0^[n/3] (-1)^k nC3k is a multiple of 3 for n > 2. (nCm is the binomial coefficient)

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40th Polish Mathematical Olympiad Problems 1989

40th Polish Mathematical Olympiad Problems 1989

A1.  An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break.

A2.  k₁, k₂, k₃ are three circles. k₂ and k₃ touch externally at P, k₃ and k₁ touch externally at Q, and k₁ and k₂ touch externally at R. The line PQ meets k₁ again at S, the line PR meets k₁ again at T. The line RS meets k₂ again at U, and the line QT meets k₃ again at V. Show that P, U, V are collinear.

A3.  The edges of a cube are labeled from 1 to 12. Show that there must exist at least eight triples (i, j, k) with 1 ≤ i < j < k ≤ 12 so that the edges i, j, k are consecutive edges of a path. But show that the labeling can be done so that we cannot find nine such triples.

B1.  n, k are positive integers. A₀ is the set {1, 2, ... , n}. A_i is a randomly chosen subset of A_i-1 (with each subset having equal probability). Show that the expected number of elements of A_k is n/2^k.

B2.  Three circles of radius a are drawn on the surface of a sphere of radius r. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

B3.  Show that for positive reals a, b, c, d we have ((ab + ac + ad + bc + bd + cd)/6)^1/2 ≥ ((abc + abd + acd + bcd)/4)^1/3

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39th Polish Mathematical Olympiad Problems 1988

39th Polish Mathematical Olympiad Problems 1988

A1.  The real numbers x₁, x₂, ... , x_n belong to the interval (0,1) and satisfy x₁ + x₂ + ... + x_n = m + r, where m is an integer and r ∈ [0,1). Show that x₁² + x₂² + ... + x_n² ≤ m + r².

A2.  For a permutation P = (p₁, p₂, ... , p_n) of (1, 2, ... , n) define X(P) as the number of j such that p_i < p_j for every i < j. What is the expected value of X(P) if each permutation is equally likely?

A3.  W is a polygon. W has a center of symmetry S such that if P belongs to W, then so does P', where S is the midpoint of PP'. Show that there is a parallelogram V containing W such that the midpoint of each side of V lies on the border of W.

B1.  d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d). Show that there exists x ∈ [0,d-1] such that f(x) = f(x+1).

B2.  The sequence a₁, a₂, a₃, ... is defined by a₁ = a₂ = a₃ = 1, a_n+3 = a_n+2a_n+1 + a_n. Show that for any positive integer r we can find s such that a_s is a multiple of r.

B3.  Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius 1.

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38th Polish Mathematical Olympiad Problems 1987

38th Polish Mathematical Olympiad Problems 1987

A1.  There are n ≥ 2 points in a square side 1. Show that one can label the points P₁, P₂, ... , P_n such that ∑_i=1ⁿ |P_i-1 - P_i|² ≤ 4, where we use cyclic subscripts, so that P₀ means P_n.

A2.  A regular n-gon is inscribed in a circle radius 1. Let X be the set of all arcs PQ, where P, Q are distinct vertices of the n-gon. 5 elements L₁, L₂, ... , L₅ of X are chosen at random (so two or more of the L_i can be the same). Show that the expected length of L₁ ∩ L₂ ∩ L₃ ∩ L₄ ∩ L₅ is independent of n.

A3.  w(x) is a polynomial with integral coefficients. Let p_n be the sum of the digits of the number w(n). Show that some value must occur infinitely often in the sequence p₁, p₂, p₃, ... .

B1.  Let S be the set of all tetrahedra which satisfy (1) the base has area 1, (2) the total face area is 4, and (3) the angles between the base and the other three faces are all equal. Find the element of S which has the largest volume.

B2.  Find the smallest n such that n²-n+11 is the product of four primes (not necessarily distinct).

B3.  A plane is tiled with regular hexagons of side 1. A is a fixed hexagon vertex. Find the number of paths P such that (1) one endpoint of P is A, (2) the other endpoint of P is a hexagon vertex, (3) P lies along hexagon edges, (4) P has length 60, and (5) there is no shorter path along hexagon edges from A to the other endpoint of P.

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37th Polish Mathematical Olympiad Problems 1986

37th Polish Mathematical Olympiad Problems 1986

A1.  A square side 1 is covered with m² rectangles. Show that there is a rectangle with perimeter at least 4/m.

A2.  Find the maximum possible volume of a tetrahedron which has three faces with area 1.

A3.  p is a prime and m is a non-negative integer < p-1. Show that ∑_j=1^p j^m is divisible by p.

B1.  Find all n such that there is a real polynomial f(x) of degree n such that f(x) ≥ f '(x) for all real x.

B2.  There is a chess tournament with 2n players (n > 1). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most n² matches. Conversely, show that if there are at most n² matches, then it is possible to arrange them so that we cannot find three players who all play each other.

B3.  ABC is a triangle. The feet of the perpendiculars from B and C to the angle bisector at A are K, L respectively. N is the midpoint of BC, and AM is an altitude. Show that K,L,N,M are concyclic.

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36th Polish Mathematical Olympiad Problems 1985

36th Polish Mathematical Olympiad Problems 1985

A1.  Find the largest k such that for every positive integer n we can find at least k numbers in the set {n+1, n+2, ... , n+16} which are coprime with n(n+17).

A2.  Given a square side 1 and 2n positive reals a₁, b₁, ... , a_n, b_n each ≤ 1 and satisfying ∑ a_ib_i ≥ 100. Show that the square can be covered with rectangles R_i with sides length (a_i, b_i) parallel to the square sides.

A3.  The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)³ for all real x and is continuous at x = 0. Show that |f(x)| ≤ 1 for all x.

B1.  P is a point inside the triangle ABC is a triangle. The distance of P from the lines BC, CA, AB is d_a, d_b, d_c respectively. Show that 2/(1/d_a + 1/d_b + 1/d_c) < r < (d_a + d_b + d_c)/2, where r is the inradius.

B2.  p(x,y) is a polynomial such that p(cos t, sin t) = 0 for all real t. Show that there is a polynomial q(x,y) such that p(x,y) = (x² + y² - 1) q(x,y).

B3.  There is a convex polyhedron with k faces. Show that if k/2 of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

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35th Polish Mathematical Olympiad Problems 1984

35th Polish Mathematical Olympiad Problems 1984

A1.  X is a set with n > 2 elements. Is there a function f : X → X such that the composition f ^n-1 is constant, but f ^n-2 is not constant?

A2.  Given n we define a_i,j as follows. For i, j = 1, 2, ... , n, a_i,j = 1 for j = i, and 0 for j ≠ i. For i = 1, 2, ... , n, j = n+1, ... , 2n, a_i,j = -1/n. Show that for any permutation p of (1, 2, ... , 2n) we have ∑_i=1ⁿ |∑_k=1ⁿ a_i,p(k) | ≥ n/2.

A3.  W is a regular octahedron with center O. P is a plane through the center O. K(O, r₁) and K(O, r₂) are circles center O and radii r₁, r₂ such that K(O, r₁) ⊆ P∩W ⊆ K(O, r₂). Show that r₁/r₂ ≤ (√3)/2.

B1.  We throw a coin n times and record the results as the sequence α₁, α₂, ... , α_n, using 1 for head, 2 for tail. Let β_j = α₁ + α₂ + ... + α_j and let p(n) be the probability that the sequence β₁, β₂, ... , β_n includes the value n. Find p(n) in terms of p(n-1) and p(n-2).

B2.  Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon with side 1. Show that no vertex of the hexagon is covered by two or more disks.

B3.  There are 1025 cities, P₁, ... , P₁₀₂₅ and ten airlines A₁, ... , A₁₀, which connect some of the cities. Given any two cities there is at least one airline which has a direct flight between them. Show that there is an airline which can offer a round trip with an odd number of flights.

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