Olympiad problems

2004 Iran Team Selection Test, 1

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

1997 Greece Junior Math Olympiad, 3

Tags: combinatorics

Establish if we can rewrite the numbers $1,2,3,4,5,6,7,8,9,10$ in a row in such a way that: (a) The sum of any three consecutive numbers (in the new order) does not exceed $16$. (b) The sum of any three consecutive numbers (in the new order) does not exceed $15$.

2008 iTest Tournament of Champions, 4

Tags:

If $m$ is a positive integer, let $S_m$ be the set of rational numbers in reduced form with denominator at most $m$. Let $f(m)$ be the sum of the numerator and denominator of the element of $S_m$ closest to $e$ (Euler's constant). Given that $f(2007) = 3722$, find the remainder when $f(1000)$ is divided by $2008$.

2011 Korea National Olympiad, 4

Tags: combinatorics

Let $k,n$ be positive integers. There are $kn$ points $P_1, P_2, \cdots, P_{kn}$ on a circle. We can color each points with one of color $ c_1, c_2, \cdots , c_k $. In how many ways we can color the points satisfying the following conditions? (a) Each color is used $ n $ times. (b) $ \forall i \not = j $, if $ P_a $ and $ P_b $ is colored with color $ c_i $ , and $ P_c $ and $ P_d $ is colored with color $ c_j $, then the segment $ P_a P_b $ and segment $ P_c P_d $ doesn't meet together.

2014 Romania National Olympiad, 4

Tags: number theory , prime number , group theory , abstract algebra

Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $ [b]a)[/b] Prove that the order of $ G $ is a power of $ p. $ [b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $

LMT Speed Rounds, 4

Tags: speed , combi

The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even. [i]Proposed by Muztaba Syed and Derek Zhao[/i] [hide=Solution] [i]Solution. [/i]$\boxed{\dfrac{1}{3}}$ Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the answer is $\boxed{\dfrac{1}{3}}$.[/hide]

Bangladesh Mathematical Olympiad 2020 Final, #6

Tags: analytic geometry , geometry

Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$[b]ABP =[/b] $\angle$[b]ADP[/b]. What is the radius of the circumcircle of $ABCD$?

2006 JBMO ShortLists, 7

Tags: number theory

Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.

2005 Tournament of Towns, 1

Tags: geometry

In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$. [i](3 points)[/i]

2021 MOAA, 18

Tags:

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

2015 Postal Coaching, 2

Tags: number theory , bounded

Prove that there exists a real number $C > 1$ with the following property. Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.

2007 Tournament Of Towns, 3

Tags: circumcircle , geometry

$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?

2025 Ukraine National Mathematical Olympiad, 9.6

Tags: number theory

The sum of $10$ positive integer numbers is equal to $300$. The product of their factorials is a perfect tenth power of some positive integer. Prove that all $10$ numbers are equal to each other. [i]Proposed by Pavlo Protsenko[/i]

2017 AMC 10, 23

Tags: counting

How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive? $\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$

2001 Pan African, 2

Tags:

Find the value of the sum: \[ \sum_{i=1}^{2001} [\sqrt{i}] \] where $[ {x} ]$ denotes the greatest integer which does not exceed $x$.

2002 Estonia National Olympiad, 4

Tags: combinatorics

Mary writes $5$ numbers on the blackboard. On each step John replaces one of the numbers on the blackboard by the number $x + y - z$, where $x, y$ and $z$ are three of the four other numbers on the blackboard. Can John make all five numbers on the blackboard equal, regardless of the numbers initially written by Mary?

2022 Bulgaria National Olympiad, 3

Tags: number theory , coprime numbers , perfect square

Let $x>y>2022$ be positive integers such that $xy+x+y$ is a perfect square. Is it possible for every positive integer $z$ from the interval $[x+3y+1,3x+y+1]$ the numbers $x+y+z$ and $x^2+xy+y^2$ not to be coprime?

2006 Romania Team Selection Test, 4

Tags: inequalities , calculus , algebra , three variable inequality

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]

2009 Paraguay Mathematical Olympiad, 3

Tags: number theory

Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.

2008 Miklós Schweitzer, 10

Tags: vector , function

Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$, and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$? (translated by Miklós Maróti)

2025 Azerbaijan Senior NMO, 2

Tags: algebra

Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$

2024 India IMOTC, 19

Tags: algebra , functional equation

Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\ [color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\] [color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]} f(X) = 2^{n+1}-1.\] (Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\ [i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]

2006 Iran Team Selection Test, 1

Tags: algebra , polynomial , combinatorial geometry , ring theory , combinatorics

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

2003 AIME Problems, 2

Tags:

Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$

1975 Dutch Mathematical Olympiad, 1

Tags: algebra

Are the following statements true? $x^7 \in Q \land x^{12} \in Q \Rightarrow x \in Q$, and $x^9 \in \land x^{12} \in Q \Rightarrow x \in Q$.