Found problems: 85335
2000 Vietnam Team Selection Test, 3
Two players alternately replace the stars in the expression
\[*x^{2000}+*x^{1999}+...+*x+1 \]
by real numbers. The player who makes the last move loses if the resulting polynomial has a real root $t$ with $|t| < 1$, and wins otherwise. Give a winning strategy for one of the players.
2020 Portugal MO, 4
Determine the fractions of a fraction of the form $\frac{1}{ab}$ where $a,b$ are prime natural numbers such that $0 < a < b \le 200$ and $a + b > 200$
2015 Swedish Mathematical Competition, 2
Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.
1969 IMO Shortlist, 35
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
2016 Taiwan TST Round 1, 4
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2014 Thailand TSTST, 2
In a triangle $ABC$, the incircle with incenter $I$ is tangent to $BC$ at $A_1, CA$ at $B_1$, and $AB$ at $C_1$. Denote the intersection of $AA_1$ and $BB_1$ by $G$, the intersection of $AC$ and $A_1C_1$ by $X$, and the intersection of $BC$ and $B_1C_1$ by $Y$ . Prove that $IG \perp XY$ .
1983 IMO Shortlist, 5
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.
1998 All-Russian Olympiad Regional Round, 11.6
A polygon with sides running along the sides of the squares was cut out of an endless chessboard. A segment of the perimeter of a polygon is called black if the polygon adjacent to it from the inside is which cell is black, respectively white if the cell is white. Let $A$ be the number of black segments on the perimeter, and $B$ be the number of white ones, Let the polygon consist of $a$ black and $b$ white cells. Prove that $A-B = 4(a -b)$.
2016 USAJMO, 1
The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively.
Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.
2024 Brazil Cono Sur TST, 2
For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.
2020/2021 Tournament of Towns, P1
The number $2021 = 43 \cdot 47$ is composite. Prove that if we insert any number of digits “8” between 20 and 21 then the number remains composite.
[i]Mikhail Evdikomov[/i]
1985 Traian Lălescu, 2.2
Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that:
$$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$
2006 Belarusian National Olympiad, 6
An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum?
(V. Lebed)
[img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]
1997 Tournament Of Towns, (531) 3
In a chess tournament, each of $2n$ players plays every other player once in each of two rounds. A win is worth $1$ point, a draw is worth $\frac12$ point and a loss is worth nothing. Prove that if for every player, the total score in the first round differs from that in the second round by at least n points, then this difference is exactly n points for every player.
(B Frenkin)
2006 Bulgaria Team Selection Test, 2
a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$.
b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$.
[i]Nikolai Nikolov, Emil Kolev[/i]
2021 BMT, 13
A six-sided die is rolled four times. What is the probability that the minimum value of the four rolls is $4$?
2022 Bulgarian Autumn Math Competition, Problem 9.1
Given is the equation:
\[x^2+mx+2022=0\]
a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers
b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers
2013 Saudi Arabia IMO TST, 2
Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that
(a) $a_1 + a_2 + .. + a_n \ge n^2$ and
(b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$
2025 EGMO, 2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called [i]central[/i] if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.
\\Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
1967 IMO Longlists, 50
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2006 Stanford Mathematics Tournament, 12
Find the total number of $k$-tuples $(n_1,n_2,...,n_k)$ of positive integers so that $n_{i+1}\ge n_i$ for each $i$, and $k$ regular polygons with numbers of sides $n_1,n_2,...,n_k$ respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is $360^{\circ}$.
1964 Poland - Second Round, 6
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.
2011 NIMO Summer Contest, 4
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality
\[
1 < a < b+2 < 10.
\]
[i]Proposed by Lewis Chen
[/i]
2010 Dutch IMO TST, 5
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
2006 District Olympiad, 2
Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.