Jan 31, 2012

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Hungarian Mathematical Olympiad 2001/02. Final Round. Grades 11 and 12 – technical schools. 1. The perpendicular from vertex A of the rectangle ABCD to ...
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Hungarian Mathematical Olympiad 2005/06. Final Round. High Schools. 1. For each nonnegative integer n, define t(0) = t(1) = 0, t(2) = 1 and for n > 2 define f(n) ...
3. [PDF]

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Hungarian Mathematical Olympiad 2000/01. Final Round. Grades 11 and 12. 1. Let S denote the number of 77-element subsets of H = {1, 2,..., 2001} ...
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Hungarian Mathematical Olympiad 1997/98. Final Round. Grades 11 and 12. 1. Prove that the arithmetic mean of the numbers 2sin2◦, 4 sin 4◦, 6 sin 6◦, ...
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Hungarian Mathematical Olympiad 1999/2000. Final Round. Grades 11 and 12. 1. Let H be a set of 2000 nonzero real numbers. How many negative elements ...
6. [PDF]

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Hungarian Mathematical Olympiad 1998/99. Final Round. Grades 11 and 12. 1. Find all solutions 0 < x, y, z ≤ 1 of the equation x. 1 + y + zx. + y. 1 + z + xy. + ...
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Eötvös Mathematical Competition 1905

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Eötvös Mathematical Competition 1905. 1. Find the necessary and sufficient conditions on positive integers n, p for the system of equations x + py = n, x + y = pz ...
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Eötvös Mathematical Competition 1897

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Eötvös Mathematical Competition 1897. 1. If α,β,γ are the angles of a right triangle, prove the relation: sin α sin β sin(α − β) + sinβ sinγ sin(β − γ) + sin γ sin α sin(γ ...
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Eötvös Mathematical Competition 1900

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Eötvös Mathematical Competition 1900. 1. Let a, b, c, d be fixed integers with d not divisible by 5. Assume that there is an integer m for which am3 + bm2 + cm+ d ...
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Kürschák Mathematical Competition 1982

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Kürschák Mathematical Competition 1982. 1. For any five points A, B, P, Q, R in a plane, prove that. AB + PQ + QR + RP ≤ AP + AQ + AR + BP + BQ + BR. 2. ...
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Eötvös Mathematical Competition 1903

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Eötvös Mathematical Competition 1903. 1. Suppose that 2p. − 1 is a prime number. Prove that the sum of all positive divisors of n = 2p−1(2p. − 1) (excluding n) is ...
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Eötvös Mathematical Competition 1899

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Eötvös Mathematical Competition 1899. 1. The points A0,A1,A2,A3,A4 divide a unit circle into five equal parts. Prove that the chords A0A1 and A0A2 satisfy ...
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Kürschák Mathematical Competition 1981

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Kürschák Mathematical Competition 1981. 1. The points of space are colored with five colors, all colors being used. Prove that some plane contains four points ...
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Eötvös Mathematical Competition 1896

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Eötvös Mathematical Competition 1896. 1. If k is the number of distinct prime divisors of a natural number n, prove that log n ≥ k log 2. 2. Prove that the ...
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Eötvös Mathematical Competition 1902

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Eötvös Mathematical Competition 1902. 1. Consider an arbitrary quadratic polynomial Q(x) = Ax2 + Bx + C. (a) Prove that Q(x) can be written in the form. Q(x) = k ...
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Kürschák Mathematical Competition 1986

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Kürschák Mathematical Competition 1986. 1. The convex (n + 1)-gon P0P1 ...Pn is partitioned into triangles by n − 2 nonintersecting diagonals. Prove that the ...
17. Eötvös Mathematical Competition 1909

Eötvös Mathematical Competition 1909. 1. Consider any three consecutive natural numbers. Prove that the cube of the largest number cannot be the sum of the ...
18. Eötvös Mathematical Competition 1939

Eötvös Mathematical Competition 1939. 1. Let a1,a2,b1,b2,c1,c2 be real numbers for which a1a2 > 0, a1c1 ≥ b2. 1 and a2c2 > b2. 2. Prove that. (a1 + a2)(c1 + ...
19. Eötvös Mathematical Competition 1936

Eötvös Mathematical Competition 1936. 1. Prove that for all positive integers n,. 1. 1 · 2. +. 1. 3 · 4. + ···. 1. (2n − 1)2n. = 1 n + 1. +. 1 n + 2. + ··· +. 1. 2n. 2. A point S ...
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Kürschák Mathematical Competition 1984

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Kürschák Mathematical Competition 1984. 1. Rational numbers x, y and z satisfy the equation x3 + 3y3 + 9z3. − 9xyz = 0. Prove that x = y = z = 0. 2. ...
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Eötvös Mathematical Competition 1911

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Eötvös Mathematical Competition 1911. 1. Show that, if the real numbers a, b, c, A,B,C satisfy. aC − 2bB + cA = 0 and ac − b2 > 0, then AC − B2 < 0. 2. Let Q be ...
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Eötvös Mathematical Competition 1938

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Eötvös Mathematical Competition 1938. 1. Prove that an integer n has a representation as a sum of two squares if and only if so does 2n. 2. Prove that for all ...
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Eötvös Mathematical Competition 1894

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Eötvös Mathematical Competition 1894. 1. Let x and y be integers. Prove that one of the expressions. 2x + 3y and 9x + 5y is divisible by 17 if and only if so is the ...
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Eötvös Mathematical Competition 1937

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Eötvös Mathematical Competition 1937. 1. Let a1,a2,...,an be positive integers and k be an integer greater than the sum of the ai's. Prove that a1!a2! ··· an! < k!. 2. ...
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Eötvös Mathematical Competition 1904

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Eötvös Mathematical Competition 1904. 1. Prove that if an inscribed pentagon has equal angles then its sides are equal. 2. If a is a natural number, show that the ...
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Eötvös Mathematical Competition 1898

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Eötvös Mathematical Competition 1898. 1. Determine all positive integers n for which 2n + 1 is divisible by 3. 2. Prove the following statement: If two triangles ...
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Eötvös Mathematical Competition 1907

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Eötvös Mathematical Competition 1907. 1. If p and q are odd integers, prove that the equation x2 + 2px +2q = 0 has no rational roots. 2. Let P be a point inside ...
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Eötvös Mathematical Competition 1895

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Eötvös Mathematical Competition 1895. 1. Prove that there are exactly 2 (2n−1. − 1) ways of dealing n cards to two persons. (The persons may receive unequal ...
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Eötvös Mathematical Competition 1906

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Eötvös Mathematical Competition 1906. 1. Prove that if tan α. 2 is rational (or undefined) then so are cosα and sin α;. Conversely, if cosα and sinα are rational ...
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Eötvös Mathematical Competition 1901

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Eötvös Mathematical Competition 1901. 1. Prove that, for any positive integer n, 1n + 2n + 3n + 4n is divisible by 5 if and only if n is not divisible by 4. 2. If u = cot ...
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Eötvös Mathematical Competition 1910

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Eötvös Mathematical Competition 1910. 1. If real numbers a, b, c satisfy a2 + b2 + c2 = 1, prove the inequalities. −. 1. 2. ≤ ab + bc + ca ≤ 1. 2. Let a, b, c, d and u ...
32. [PDF]

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Hungarian Mathematical Olympiad 2002/03. Final Round. Category 1. 1. Solve the equation (x2 + 5x + 1)(x2 + 4x) = 20(x + 1)2 in R. 2. The medians sa,sb,sc and ...
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Kürschák Mathematical Competition 1983

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Kürschák Mathematical Competition 1983. 1. A cube of integer edge lengths is given in space so that all four vertices of one of the faces are lattice points. ...
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Kürschák Mathematical Competition 1985

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Kürschák Mathematical Competition 1985. 1. Writing down the first 4 rows of the Pascal triangle in the usual way and then adding up the numbers in vertical ...
35. Eötvös Mathematical Competition 1908

Eötvös Mathematical Competition 1908. 1. If a and b are odd integers, prove that a3. −b. 3 is divisible by 2n if and only if so is a − b. 2. Let n > 2 be an integer. ...