Olympiad problems

2014 Math Hour Olympiad, 8-10.2

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A complete set of the Encyclopedia of Mathematics has $10$ volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.

2016 ASDAN Math Tournament, 10

Tags: algebra test

Let $a_1,a_2,\dots$ be a sequence of real numbers satisfying $$\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}.$$ Given that $a_1=-1$ and $a_2=-\tfrac{1}{2}$, find the value of $\tfrac{a_9}{a_{20}}$.

2012 Belarus Team Selection Test, 3

Tags: algebra , inequalities , polynomial

Given a polynomial $P(x)$ with positive real coefficients. Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$. (K. Gorodnin)

2016 239 Open Mathematical Olympiad, 4

Tags: number theory , sequence

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$ Prove that $p_n$ and $q_m$ are coprime for any m and n.

2011 Irish Math Olympiad, 2

Tags: combinatorial

In a tournament with $n$ players, $n$ < 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?

2004 Switzerland - Final Round, 3

Tags: number theory

Let $p$ be an odd prime number. Find all natural numbers $k$ such that $$\sqrt{k^2 - pk}$$ is a positive integer.

2000 Tuymaada Olympiad, 1

Tags: number theory

Let $d(n)$ denote the number of positive divisors of $n$ and let $e(n)=\left[2000\over n\right]$ for positive integer $n$. Prove that \[d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).\]

1999 Tuymaada Olympiad, 4

Tags: inequalities

Prove the inequality \[ {x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1, \] where $2 < x, y, z < 4.$ [i]Proposed by A. Golovanov[/i]

2016 Harvard-MIT Mathematics Tournament, 5

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Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of $10$ centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of $4$ centimeters. Given that the total volume of the snowballs that Anderson constructs cannot exceed the volume of the snowball that Patrick threw, how many snowballs can Anderson construct?

2009 Germany Team Selection Test, 2

Tags: function , algebra , functional inequality

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2013 Kosovo National Mathematical Olympiad, 5

Tags: geometry , perimeter , inequalities , triangle inequality

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

2021 JHMT HS, 5

Tags: geometry

Let $\mathcal{S}$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $xy > 0$ and $x^2 + y^2 + 2x + 4y \leq 2021.$ The total area of $\mathcal{S}$ can be written in the form $a\pi + b,$ where $a$ and $b$ are integers. Compute $a + b.$

2009 Sharygin Geometry Olympiad, 15

Tags: trigonometry , geometry

Given a circle and a point $ C$ not lying on this circle. Consider all triangles $ ABC$ such that points $ A$ and $ B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

2007 JBMO Shortlist, 4

Tags: algebra , positive integer

Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

2021 AMC 10 Fall, 3

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What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

2017 China Second Round Olympiad, 2

Tags: algebra , sequence

Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{ \begin{array}{lcr} a_n+n,\quad a_n\le n, \\ a_n-n,\quad a_n>n, \end{array} \right. \quad n=1,2,\cdots.$ Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.

2019 Dutch BxMO TST, 2

Tags: geometry , angle bisector , circumcircle

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

2017 ITAMO, 5

Tags: number theory , algebra , ramsey theory , combinatorics

Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.

2021 Malaysia IMONST 1, 16

Tags: geometry , octagon , square , area

Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

1966 IMO Shortlist, 48

Tags: algebra , equation , number theory , root , parametric equation

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2010 China Team Selection Test, 2

Tags: induction , number theory , greatest common divisor , algebra

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

1954 AMC 12/AHSME, 32

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The factors of $ x^4\plus{}64$ are: $ \textbf{(A)}\ (x^2\plus{}8)^2 \qquad \textbf{(B)}\ (x^2\plus{}8)(x^2\minus{}8) \qquad \textbf{(C)}\ (x^2\plus{}2x\plus{}4)(x^2\minus{}8x\plus{}16) \\ \textbf{(D)}\ (x^2\minus{}4x\plus{}8)(x^2\minus{}4x\minus{}8) \qquad \textbf{(E)}\ (x^2\minus{}4x\plus{}8)(x^2\plus{}4x\plus{}8)$

2004 Alexandru Myller, 3

Tags: abstract algebra , ring theory , nilpotence , superior algebra

Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

1979 Czech And Slovak Olympiad IIIA, 4

Tags: algebra , inequalities

Let $n$ be any natural number. Find all $n$-tuples of real numbers $x_1\le x_2\le ... \le x_n$, for which holds $$\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.$$

1959 AMC 12/AHSME, 27

Tags: quadratic , vieta , algebra , polynomial , quadratic formula , quadratic equation

Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$? $ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$ $\textbf{(B)}\ \text{The discriminant is 9}\qquad$ $\textbf{(C)}\ \text{The roots are imaginary}\qquad$ $\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$ $\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $