Found problems: 85335
2018 Stanford Mathematics Tournament, 2
Consider a game played on the integers in the closed interval $[1, n]$. The game begins with some tokens placed in $[1, n]$. At each turn, tokens are added or removed from$ [1, n]$ using the following rule: For each integer $k \in [1, n]$, if exactly one of $k - 1$ and $k + 1$ has a token, place a token at $k$ for the next turn, otherwise leave k blank for the next turn.
We call a position [i]static [/i] if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.
2018 Brazil National Olympiad, 1
We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.
2019 Bosnia and Herzegovina Junior BMO TST, 1
Let $x,y,z$ be real numbers ( $x \ne y$, $y\ne z$, $x\ne z$) different from $0$. If $\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}$, prove that the following relation holds:
$$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$
2022 Balkan MO Shortlist, C5
Given is a cube of side length $2021$. In how many different ways is it possible to add somewhere on the boundary of this cube a $1\times 1\times 1$ cube in such a way that the new shape can be filled in with $1\times 1\times k$ shapes, for some natural number $k$, $k\geq 2$?
2013 Kazakhstan National Olympiad, 1
Find maximum value of
$|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.
2016 Tournament Of Towns, 1
$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies?
[i](N. Chernyatevya)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
1961 IMO Shortlist, 2
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?
1996 IberoAmerican, 1
Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that:
$a < b \leq n$,
$a+b>n$.
Show that for each $n$, the sum of all this fractions are $\frac12$.
2017 Bundeswettbewerb Mathematik, 1
For which integers $n \geq 4$ is the following procedure possible? Remove one number of the integers $1,2,3,\dots,n+1$ and arrange them in a sequence $a_1,a_2,\dots,a_n$ such that of the $n$ numbers \[ |a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1| \] no two are equal.
2014 AIME Problems, 7
Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]
1999 Switzerland Team Selection Test, 3
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.
2023-IMOC, G2
$P$ is a point inside $\triangle ABC$. $AP, BP, CP$ intersects $BC, CA, AB$ at $D, E, F$, respectively. $AD$ meets $(ABC)$ again at $D_1$. $S$ is a point on $(ABC)$. Lines $AS$, $EF$ intersect at $T$, lines $TP, BC$ intersect at $K$, and $KD_1$ meets $(ABC)$ again at $X$. Prove that $S, D, X$ are colinear.
LMT Guts Rounds, 2020 F24
In the Oxtingnle math team, there are $5$ students, numbered $1$ to $5$, all of which either always tell the truth or always lie. When Marpeh asks the team about how they did in a $10$ question competition, each student $i$ makes $5$ separate statements (so either they are all false or all true): "I got problems $i+1$ to $2i$, inclusive, wrong", and then "Student $j$ got both problems $i$ and $2i$ correct" for all $j \neq i$. What is the most problems the team could have gotten correctly?
[i]Proposed by Jeff Lin[/i]
1996 Canadian Open Math Challenge, 8
Determine all pairs of integers $(x,y)$ which satisfy the equation
\[ 6x^2-3xy-13x+5y = -11 \]
2019 BMT Spring, 2
A set of points in the plane is called [i]full[/i] if every triple of points in the set are the vertices of a non-obtuse triangle. What is the largest size of a full set?
2009 China National Olympiad, 2
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$
2000 Moldova National Olympiad, Problem 3
For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.
2011 Tournament of Towns, 3
(a) Does there exist an in nite triangular beam such that two of its cross-sections are similar but not congruent triangles?
(b) Does there exist an in nite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?
2015 ASDAN Math Tournament, 1
A rectangle $ABCD$ is split into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints are on the sides of $ABCD$. If the smallest three rectangles have areas of $48$, $18$, and $12$, what is the area of $ABCD$?
2021 Azerbaijan IZhO TST, 1
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
2019 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.
2023 Baltic Way, 3
Denote a set of equations in the real numbers with variables $x_1, x_2, x_3 \in \mathbb{R}$ Flensburgian if there exists an $i \in \{1, 2, 3\}$ such that every solution of the set of equations where all the variables are pairwise different, satisfies $x_i>x_j$ for all $j \neq i$.
Find all positive integers $n \geq 2$, such that the following set of two equations $a^n+b=a$ and $c^{n+1}+b^2=ab$ in three real variables $a,b,c$ is Flensburgian.
2020 Greece JBMO TST, 4
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$.
PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3 [/url]
2018 Iran MO (1st Round), 23
Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the needle leg could be positioned at any angle with respect to the paper. Let $n$ be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to $n$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 20$
1957 Moscow Mathematical Olympiad, 372
Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.