Olympiad problems

2019 International Zhautykov OIympiad, 2

Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

Russian TST 2022, P2

Tags: algebra , Sequence , combinatorics , izho

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

2025 International Zhautykov Olympiad, 1

Tags: algebra , inequalities , izho , inequalities proposed

Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]

2024 International Zhautykov Olympiad, 1

Tags: combinatorics , izho

In an alphabet of $n$ letters, is $syllable$ is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered $indecent$. A $word$ is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of indecent syllables for which infinite words do not exist.

2021 XVII International Zhautykov Olympiad, #4

Tags: IZHO 2021 , izho , geometric inequality , geometry , inequalities

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

2019 International Zhautykov OIympiad, 1

Tags: izho , combinatorics , factorial

Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

2022 International Zhautykov Olympiad, 6

Tags: algebra , Sequence , combinatorics , izho

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

2021 XVII International Zhautykov Olympiad, #2

Tags: geometry , hexagon , izho

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

2021 International Zhautykov Olympiad, 2

Tags: geometry , hexagon , izho

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

2019 International Zhautykov OIympiad, 6

Tags: polynomial , algebra , izho

We define two types of operation on polynomial of third degree: a) switch places of the coefficients of polynomial(including zero coefficients), ex: $ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$ b) replace the polynomial $P(x)$ with $P(x+1)$ If limitless amount of operations is allowed, is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?

2021 International Zhautykov Olympiad, 4

Tags: IZHO 2021 , izho , geometric inequality , geometry , inequalities

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

2025 International Zhautykov Olympiad, 4

Tags: izho , number theory , combinatorics

Vaysha has a board with $999$ consecutive numbers written and $999$ labels of the form [i]"This number is [b]not [/b]divisible by $i$"[/i], for $i \in \{ 2,3, \dots ,1000 \} $. She places each label next to a number on the board, so that each number has exactly one label. For each true statement on the stickers, Vaysha gets a piece of candy. How many pieces of candy can Vaysha guarantee to win, regardless of the numbers written on the board, if she plays optimally?