Found problems: 85335
2021 Sharygin Geometry Olympiad, 8
Let $ABC$ be an isosceles triangle ($AB=BC$) and $\ell$ be a ray from $B$. Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$. Prove that $\angle PAQ=\angle PCQ$.
Kvant 2019, M2569
Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.
1969 All Soviet Union Mathematical Olympiad, 124
Given a pentagon with all equal sides.
a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle. (side $AB$ is seen from the point $C$ inside an arbitrary angle that is greater or equal than $\angle ACB$)
b) Prove that the circles constructed on its sides as on the diameters cannot cover the pentagon entirely.
2016 Romania National Olympiad, 2
Let $A$ be a ring and let $D$ be the set of its non-invertible elements. If $a^2=0$ for any $a \in D,$ prove that:
[b]a)[/b] $axa=0$ for all $a \in D$ and $x \in A$;
[b]b)[/b] if $D$ is a finite set with at least two elements, then there is $a \in D,$ $a \neq 0,$ such that $ab=ba=0,$ for every $b \in D.$
[i]Ioan Băetu[/i]
2019 India PRMO, 27
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?
[center][img]https://i.imgur.com/T1Y3zwR.png[/img][/center]
2007 Alexandru Myller, 1
Solve $ x^3-y^3=2xy+7 $ in integers.
Novosibirsk Oral Geo Oly VII, 2019.2
Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.
[url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]
2011 India IMO Training Camp, 2
Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.
2016 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
2019 Thailand TST, 1
Let $n$ be a positive integer. Let $S$ be a set of $n$ positive integers such that the greatest common divisors of all nonempty sets of $S$ are distinct. Determine the smallest possible number of distinct prime divisors of the product of the elements of $S$.
2002 Manhattan Mathematical Olympiad, 3
The product $1 \cdot 2 \cdot \ldots \cdot n$ is denoted by $n!$ and called [i]n-factorial[/i]. Prove that the product
\[ 1!2!3!\ldots 49!51! \ldots 100! \]
(the factor $50!$ is missing)
is the square of an integer number.
PEN S Problems, 17
Determine the maximum value of $m^{2}+n^{2}$, where $m$ and $n$ are integers satisfying $m,n\in \{1,2,...,1981\}$ and $(n^{2}-mn-m^{2})^{2}=1.$
2016 Bangladesh Mathematical Olympiad, 8
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.
1976 AMC 12/AHSME, 6
If $c$ is a real number and the negative of one of the solutions of $x^2-3x+c=0$ is a solution of $x^2+3x-c=0$, then the solutions of $x^2-3x+c=0$ are
$\textbf{(A) }1,~2\qquad\textbf{(B) }-1,~-2\qquad\textbf{(C) }0,~3\qquad\textbf{(D) }0,~-3\qquad \textbf{(E) }\frac{3}{2},~\frac{3}{2}$
1941 Moscow Mathematical Olympiad, 089
Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.
2010 Thailand Mathematical Olympiad, 5
Determine all functions $f : R \times R \to R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$
2014 Harvard-MIT Mathematics Tournament, 10
An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.
Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.
2020 China National Olympiad, 5
Given any positive integer $c$, denote $p(c)$ as the largest prime factor of $c$. A sequence $\{a_n\}$ of positive integers satisfies $a_1>1$ and $a_{n+1}=a_n+p(a_n)$ for all $n\ge 1$. Prove that there must exist at least one perfect square in sequence $\{a_n\}$.
2008 AIME Problems, 9
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.
2011 Math Prize For Girls Problems, 1
If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $\left| m - n \right|$ ?
2016 Bundeswettbewerb Mathematik, 4
There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard.
Prove that there are at least two children in this class that have the same forename and surname.
1961 AMC 12/AHSME, 7
When simplified, the third term in the expansion of $\left(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2}\right)^6$ is:
${{ \textbf{(A)}\ \frac{15}{x}\qquad\textbf{(B)}\ -\frac{15}{x}\qquad\textbf{(C)}\ -\frac{6x^2}{a^9} \qquad\textbf{(D)}\ \frac{20}{a^3} }\qquad\textbf{(E)}\ -\frac{20}{a^3} } $
2008 Germany Team Selection Test, 2
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]
2018 Czech-Polish-Slovak Match, 2
Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$.
[i]Proposed by Patrik Bak, Slovakia[/i]
2020 Dutch IMO TST, 3
For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically?
Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.