Olympiad problems

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

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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 $x^9 \in \land x^{12} \in Q \Rightarrow x \in Q$.

2010 Canada National Olympiad, 4

Tags: abstract algebra , induction , combinatorics

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

1959 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry

Consider a piece of material in the shape of a right circular conical frustum with radii $R,r,R>r$. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii $R',r'$ of the cavity. Decide for which ratio $R/r$ the problem has a solution. [img]https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png[/img]

1995 Irish Math Olympiad, 3

Tags: algebra

Let $S$ be the square consisting of all pints $(x,y)$ in the plane with $0\le x,y\le 1$. For each real number $t$ with $0<t<1$, let $C_t$ denote the set of all points $(x,y)\in S$ such that $(x,y)$ is on or above the line joining $(t,0)$ to $(0,1-t)$. Prove that the points common to all $C_t$ are those points in $S$ that are on or above the curve $\sqrt{x}+\sqrt{y}=1$.

2009 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , 3d geometry

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

2024 ITAMO, 3

Tags: egyptian fractions , reciprocal sum , number theory

A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\] (a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

2010 Greece National Olympiad, 2

Tags: inequalities

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

1991 Arnold's Trivium, 2

Tags: limit , trigonometry

Find the limit \[\lim_{x\to0}\frac{\sin \tan x-\tan\sin x}{\arcsin\arctan x-\arctan\arcsin x}\]

2021 AIME Problems, 12

Tags:

Let $A_1A_2A_3...A_{12}$ be a dodecagon (12-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1986 Brazil National Olympiad, 5

Tags: combinatorics , number theory , maximum , chessboard

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

2017 NIMO Problems, 8

Tags:

Let $ABC$ be a triangle with $BC=49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\dfrac{AB \cdot AC}{-BC+AB+AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

2021 Latvia Baltic Way TST, P3

Tags: algebra , system of equations

Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

1995 Tournament Of Towns, (467) 1

Tags: geometry

Prove that inside any acute-angled triangle, there exists a point $P$ such that the feet of the perpendiculars dropped from $P$ to the sides of the triangle are the vertices of an equilateral triangle. (NB Vassiliev)