This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 36

2019 Moldova Team Selection Test, 2
Tags: trigonometry , easy , geometry
Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$. (A natural number is a positive integer)

2024 All-Russian Olympiad Regional Round, 11.6
Tags: algebra , easy , weight
Teacher has 100 weights with masses $1$ g, $2$ g, $\dots$, $100$ g. He wants to give 30 weights to Petya and 30 weights to Vasya so that no 11 Petya's weights have the same total mass as some 12 Vasya's weights, and no 11 Vasya's weights have the same total mass as some 12 Petya's weights. Can the teacher do that?

2018 USAJMO, 3
Tags: orthocenter , easy , geometry
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

2009 Dutch IMO TST, 1
Tags: induction , number theory , easy
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

2019 Moldova Team Selection Test, 3
Tags: combinatorics , easy
On the table there are written numbers $673, 674, \cdots, 2018, 2019.$ Nibab chooses arbitrarily three numbers $a,b$ and $c$, erases them and writes the number $\frac{\min(a,b,c)}{3}$, then he continues in an analogous way. After Nibab performed this operation $673$ times, on the table remained a single number $k$. Prove that $k\in(0,1).$

1998 Akdeniz University MO, 3
Tags: easy inequality , easy , inequalities
Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

2024 Bulgarian Winter Tournament, 9.1
Tags: algebra , easy
Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$ and $$(x^2+1)(y^2+1)=10xy.$$

2011 District Olympiad, 1
Tags: increasing function , algebra , easy , district olympiad , equation
Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

2020 Polish Junior MO Second Round, 1.
Tags: algebra , easy
Let $a$, $b$, $c$ be the real numbers. It is true, that $a + b$, $b + c$ and $c + a$ are three consecutive integers, in a certain order, and the smallest of them is odd. Prove that the numbers $a$, $b$, $c$ are also consecutive integers in a certain order.

2006 Baltic Way, 16
Tags: number theory , easy , modular arithmetic
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?

2003 JHMMC 8, 4
Tags: solve equation , easy , very very easy , very easy , addition , elementary contest
A number plus $4$ is $2003$. What is the number?

2024 ITAMO, 1
Tags: absolute value , easy , irrational , algebra
Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

2014 Contests, 1
Tags: circumcircle , easy , geometry
Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called[i] beautiful[/i] if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled. [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

2021 Korea - Final Round, P4
Tags: combinatorics , easy , set
There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below. (1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$ (2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$ (3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$

2016 Tuymaada Olympiad, 1
Tags: sequence , easy , algebra
The sequence $(a_n)$ is defined by $a_1=0$, $$ a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1. $$ Prove that $a_{2016}>{1\over 2}+a_{1000}$.

2006 APMO, 5
Tags: combinatorics , ez , easy
In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.

2020 Polish Junior MO First Round, 2.
Tags: easy , geometry
Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

2009 Dutch IMO TST, 1
Tags: induction , easy , number theory
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

2023 Korea - Final Round, 4
Tags: easy , number theory
Find all positive integers $n$ satisfying the following. $$2^n-1 \text{ doesn't have a prime factor larger than } 7$$

2025 Kosovo National Mathematical Olympiad`, P1
Tags: geometry , easy , pentagon
The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?

2025 Israel National Olympiad (Gillis), P1
Tags: combinatorics , easy
Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

2023 European Mathematical Cup, 1
Tags: algebra , easy
Determine all sets of real numbers $S$ such that: [list] [*] $1$ is the smallest element of $S$, [*] for all $x,y\in S$ such that $x>y$, $\sqrt{x^2-y^2}\in S$ [/list] [i]Adian Anibal Santos Sepcic[/i]

2013 Kazakhstan National Olympiad, 1
Tags: invariant , easy , modular arithmetic , number theory
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

2023 Hong Kong Team Selection Test, Problem 1
Tags: inequality , algebra , easy
Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

2013 Kazakhstan National Olympiad, 1
Tags: invariant , number theory , easy , modular arithmetic
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.