Olympiad problems

2011 District Olympiad, 1

Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

2023 Korea - Final Round, 4

Tags: easy , number theory

Find all positive integers $n$ satisfying the following. $$2^n-1 \text{ doesn't have a prime factor larger than } 7$$

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 $n=S(n)+9T(n)$.

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 $ABC$.

1993 China National Olympiad, 5

Tags: combinatorics , easy

$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)

2024 ITAMO, 1

Tags: irrational , easy , absolute value , 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 $x_{n+2}=x_n$.

2006 Baltic Way, 16

Tags: modular arithmetic , easy , number theory

Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?

2006 APMO, 5

Tags: easy , ez , combinatorics

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.

2022 BMT, 2

Tags: easy , algebra

The equation $$4^x -5 \cdot 2^{x+1} +16 = 0$$ has two integer solutions for $x.$ Find their sum.

2023 Hong Kong Team Selection Test, Problem 1

Tags: inequality , easy , algebra

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 $2$.

2022 BMT, 1

Tags: easy , algebra

Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$

2003 JHMMC 8, 4

Tags: easy , very very easy , very easy , addition , elementary contest , solve equation

A number plus $4$ is $2003$. What is the number?

2019 Moldova Team Selection Test, 2

Tags: easy , geometry , trigonometry

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)

2018 USAJMO, 3

Tags: easy , geometry , orthocenter

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 $EQ = FR$.

2020 Polish Junior MO Second Round, 1.

Tags: easy , algebra

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.

2017 District Olympiad, 1

Tags: number theory , easy

[b]a)[/b] Let $ m,n,p\in\mathbb{Z}_{\ge 0} $ such that $ m>n $ and $ \sqrt{m} -\sqrt n=p. $ Prove that $ m $ and $ n $ are perfect squares. [b]b)[/b] Find the numbers of four digits $ \overline{abcd} $ that satisfy the equation: $$ \sqrt {\overline{abcd} } -\sqrt{\overline{acd}} =\overline{bb} . $$

2014 Contests, 1

Tags: geometry , circumcircle , easy

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]

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 $a_{2016}>{1\over 2}+a_{1000}$.

2019 Moldova Team Selection Test, 1

Tags: number theory , easy

Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$. (A natural number is a positive integer)

2025 Kosovo National Mathematical Olympiad`, P1

Tags: pentagon , easy , geometry

The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?

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 $2013^3$.

2013 India Regional Mathematical Olympiad, 3

Tags: easy , nice , number theory

Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$.

2014 International Zhautykov Olympiad, 1

Tags: geometry , circumcircle , easy

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]

2020 Polish Junior MO First Round, 1.

Tags: number theory , digit , easy

Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.

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 $n=S(n)+9T(n)$.