Olympiad problems

2024 All-Russian Olympiad Regional Round, 11.6

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?

1998 Akdeniz University MO, 3

Tags: inequalities , easy inequality , easy

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

2021 Austrian Junior Regional Competition, 1

Tags: algebra , sum , cool , easy

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

2021 Austrian MO Beginners' Competition, 1

Tags: sum , cool , easy , algebra

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

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 $k\in(0,1).$

2001 APMO, 1

Tags: number theory , induction , 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 $n=S(n)+9T(n)$.

2023 European Mathematical Cup, 1

Tags: easy , algebra

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]

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 $B_i\subseteq A_k\cup A_j$

2024 Bulgarian Winter Tournament, 9.1

Tags: algebra , easy

Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$ and $$(x^2+1)(y^2+1)=10xy.$$

2013 Kazakhstan National Olympiad, 1

Tags: modular arithmetic , invariant , easy , 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 $2013^3$.

2025 Israel National Olympiad (Gillis), P1

Tags: easy , combinatorics

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?