Olympiad problems

1972 IMO Longlists, 37

On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?

PEN A Problems, 61

Tags: divisibility theory , search

For any positive integer $n>1$, let $p(n)$ be the greatest prime divisor of $n$. Prove that there are infinitely many positive integers $n$ with \[p(n)<p(n+1)<p(n+2).\]

2009 Moldova Team Selection Test, 3

Tags: cyclic quadrilateral , circumcircle , search , geometry

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

2010 Iran MO (3rd Round), 3

Tags: polynomial , search , inequalities , algebra

prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

1982 IMO Longlists, 44

Tags: search , geometry

Let $A$ and $B$ be positions of two ships $M$ and $N$, respectively, at the moment when $N$ saw $M$ moving with constant speed $v$ following the line $Ax$. In search of help, $N$ moves with speed $kv$ ($k < 1$) along the line $By$ in order to meet $M$ as soon as possible. Denote by $C$ the point of meeting of the two ships, and set \[AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.\] Determine the angle $\angle ABC = \beta$ and time $t$ that $N$ needs in order to meet $M$.

1992 Putnam, A4

Tags: derivative , limit , trigonometry , search , function , calculus

Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$

2008 Romania Team Selection Test, 3

Tags: geometry , search , induction , combinatorics

Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that \[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]

2009 German National Olympiad, 3

Tags: search , geometry

Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively. Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle. Extension: Prove that the converse is true, too. [If this has already been posted, I humbly apologize. A quick search turned up nothing.]

2008 China Team Selection Test, 3

Tags: combinatorics , ramsey theory , arithmetic sequence , search

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

2010 Contests, 3

Tags: polynomial , search , algebra , inequalities

prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

2010 IberoAmerican Olympiad For University Students, 6

Tags: polynomial , search , number theory , algebra

Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

2010 Brazil National Olympiad, 3

Tags: greatest common divisor , modular arithmetic , search , number theory

Find all pairs $(a, b)$ of positive integers such that \[ 3^a = 2b^2 + 1. \]

1992 AMC 12/AHSME, 29

Tags: search , binomial theorem , algebra , probability , polynomial

An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even? $ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $

1963 Miklós Schweitzer, 8

Tags: real analysis , function , trigonometry , search , integration

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

1983 AIME Problems, 2

Tags: complex number , absolute value , search , function

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

2009 USA Team Selection Test, 9

Tags: function , inequalities , derivative , search , calculus , polynomial , algebra

Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]

2006 MOP Homework, 5

Tags: search , function , geometry

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180$.

1962 Miklós Schweitzer, 1

Tags: polynomial , search , advanced fields , algebra

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

2005 Turkey Team Selection Test, 1

Tags: function , number theory , search

Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

2001 AMC 8, 25

Tags: search

There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it? $ \text{(A)}\ 5724\qquad\text{(B)}\ 7245\qquad\text{(C)}\ 7254\qquad\text{(D)}\ 7425\qquad\text{(E)}\ 7542 $

2007 Germany Team Selection Test, 3

Tags: trigonometry , search , geometry , inequalities

Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove: \[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F \] When does equality occur?

2007 Moldova Team Selection Test, 2

Tags: diophantine equation , search , polynomial , number theory , algebra

Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

2005 South East Mathematical Olympiad, 5

Tags: search , power of a point , geometry , function , trigonometry

Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively. Find the value of $PB \cdot PC$.

2011 Olympic Revenge, 2

Tags: 3d geometry , search , modular arithmetic , geometry , number theory , function

Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying \[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]

2013 Finnish National High School Mathematics Competition, 5

Tags: search , quadratic , finland , modular arithmetic , number theory

Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.