Olympiad problems

2007 IMO Shortlist, 7

Let $ n$ be a positive integer. Consider \[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \} \] as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$. [i]Author: Gerhard Wöginger, Netherlands [/i]

2013 BMT Spring, 5

Tags: polynomial , algebra

Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?

Russian TST 2021, P3

Tags: combinatorics , graph theory

Given a natural number $n\geqslant 2$, find the smallest possible number of edges in a graph that has the following property: for any coloring of the vertices of the graph in $n{}$ colors, there is a vertex that has at least two neighbors of the same color as itself.

2024 HMNT, 4

Tags:

Compute the number of ways to pick a three-element subset of $$\{10^1+1, 10^2+1, 10^3+1, 10^4+1, 10^5+1, 10^6+1, 10^7+1\}$$ such that the product of the $3$ numbers in the subset has no digits besides $0$ and $1$ when written in base $10.$

2017 Yasinsky Geometry Olympiad, 3

Tags: geometry , angle bisector

The two sides of the triangle are $10$ and $15$. Prove that the length of the bisector of the angle between them is less than $12$.

2018 USAJMO, 2

Tags: inequality , homogenous

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]

2016 Peru Cono Sur TST, P3

Tags: combinatorics

Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.

2001 India National Olympiad, 6

Tags: function , algebra

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$

1999 Romania National Olympiad, 3

Tags: real analysis , mean value theorem

Let $a,b \in \mathbb{R},$ $a<b$ and $f,g:[a,b] \to \mathbb{R}$ two differentiable functions with increasing derivatives and $f'(a)>0,$ $g'(a)>0.$ Prove that there exists $c \in [a,b]$ such that $$\frac{f(b)-f(a)}{b-a} \cdot \frac{g(b)-g(a)}{b-a}=f'(c)g'(c).$$

2010 Czech-Polish-Slovak Match, 3

Tags: parallelogram , geometric transformation , reflection , inequalities , triangle inequality , geometry

Let $ABCD$ be a convex quadrilateral for which \[ AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.\] Prove that $ABCD$ is a parallelogram.

2023 Bulgaria JBMO TST, 1

Tags: algebra , system of equations , algebraic manipulations

Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.

1984 IMO Longlists, 60

Tags: algebra , logarithm , inequalities

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

KoMaL A Problems 2019/2020, A. 759

Tags: combinatorics , permutation , expected value

We choose a random permutation of $1,2,\ldots,n$ with uniform distribution. Prove that the expected value of the length of the longest increasing subsequence in the permutation is at least $\sqrt{n}.$

2005 Miklós Schweitzer, 6

Tags: linear algebra , matrix , inequalities

$SU_2(\mathbb{C})=\left\{\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix} : z,w\in\mathbb{C} , z\bar{z}+w\bar{w}=1\right\}$ A and B are 2 elements of the above matrix group and have eigenvalues $e^{i\theta_1}$ , $e^{-i\theta_1}$ and $e^{i\theta_2}$ , $e^{-i\theta_2}$respectively, where $0\leq\theta_i\leq\pi$ . Prove that if AB has eigenvalue $e^{i\theta_3}$ , then $\theta_3$ satisfies the inequality $|\theta_1-\theta_2|\leq\theta_3\leq \min\{\theta_1+\theta_2 , 2\pi-(\theta_1+\theta_2)\}$

2007 Iran Team Selection Test, 3

Tags: geometry , incenter , geometric transformation , homothety , trapezoid , analytic geometry , reflection

Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

2008 Postal Coaching, 3

Tags: complex , inequalities , algebra

Let $a$ and $b$ be two complex numbers. Prove the inequality $$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$

1950 AMC 12/AHSME, 1

Tags: ratio

If 64 is divided into three parts proportional to 2, 4, and 6, the smallest part is: $\textbf{(A)}\ 5\dfrac{1}{3} \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 10\dfrac{2}{3} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None of these answers}$

2020 Purple Comet Problems, 3

Tags: algebra

The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2006 China Second Round Olympiad, 9

Tags: ellipse , ratio , geometry , conic

Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.

2018 Regional Olympiad of Mexico Southeast, 4

Tags: number theory , infinity

For every natural $n$ let $a_n=20\dots 018$ with $n$ ceros, for example, $a_1=2018, a_3=200018, a_7=2000000018$. Prove that there are infinity values of $n$ such that $2018$ divides $a_n$

2010 Tournament Of Towns, 5

Tags: invariant

$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.

2008 Mathcenter Contest, 8

Tags: geometry , combinatorics , combinatorial geometry

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

2018 Hong Kong TST, 1

Tags: algebra , number theory

Find all positive integer(s) $n$ such that $n^2+32n+8$ is a perfect square.

2008 SDMO (Middle School), 2

Tags:

Find the sum of the first $55$ terms of the sequence $$\binom{0}{0},\quad\binom{1}{0},\quad\binom{1}{1},\quad\binom{2}{0},\quad\binom{2}{1},\quad\binom{2}{2},\quad\binom{3}{0},\quad\ldots.$$ Note: For nonnegative integers $n$ and $k$ where $0\leq k\leq n$, $$\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}.$$

1966 IMO Shortlist, 12

Tags: number theory , decimal representation , equation

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.