Found problems: 85335
2015 Polish MO Finals, 2
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
1993 AMC 12/AHSME, 17
Amy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If $t$ is the area of one of the eight triangular regions such as that between $12$ o'clock and $1$ o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between $1$ o'clock and $2$ o'clock, then $\frac{q}{t}=$
[asy]
size((80));
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--(.9,0)--(3.1,4)--(.9,4)--(3.1,0)--(2,0)--(2,4));
draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2));
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-2 \qquad\textbf{(B)}\ \frac{3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(D)}\ \sqrt{3} \qquad\textbf{(E)}\ 2 $
2009 Middle European Mathematical Olympiad, 11
Find all pairs $ (m$, $ n)$ of integers which satisfy the equation
\[ (m \plus{} n)^4 \equal{} m^2n^2 \plus{} m^2 \plus{} n^2 \plus{} 6mn.\]
2017 AIME Problems, 4
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
1992 India National Olympiad, 10
Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that \[ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} . \]
2017 Bosnia And Herzegovina - Regional Olympiad, 2
It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$
2006 Moldova National Olympiad, 11.4
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
2021 Baltic Way, 17
Distinct positive integers $a, b, c, d$ satisfy
$$\begin{cases} a \mid b^2 + c^2 + d^2,\\
b\mid a^2 + c^2 + d^2,\\
c \mid a^2 + b^2 + d^2,\\
d \mid a^2 + b^2 + c^2,\end{cases}$$
and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
2005 AMC 10, 3
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
$ \textbf{(A)}\ \frac{1}{10}\qquad
\textbf{(B)}\ \frac{1}{9}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{4}{9}\qquad
\textbf{(E)}\ \frac{5}{9}$
2004 Federal Math Competition of S&M, 2
In a triangle $ABC$, points $D$ and $E$ are taken on rays $CB$ and $CA$ respectively so that $CD=CE = \frac{AC+BC}{2}$. Let $H$ be the orthocenter of the triangle, and $P$ be the midpoint of the arc $AB$ of the circumcircle of $ABC$ not containing $C$. Prove that the line $DE$ bisects the segment $HP$.
1939 Eotvos Mathematical Competition, 3
$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.
1999 Switzerland Team Selection Test, 8
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
1954 AMC 12/AHSME, 17
The graph of the function $ f(x) \equal{} 2x^3 \minus{} 7$ goes:
$ \textbf{(A)}\ \text{up to the right and down to the left} \\
\textbf{(B)}\ \text{down to the right and up to the left} \\
\textbf{(C)}\ \text{up to the right and up to the left} \\
\textbf{(D)}\ \text{down to the right and down to the left} \\
\textbf{(E)}\ \text{none of these ways.}$
2021 Science ON grade VIII, 2
Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$.
[i] (Vlad Robu) [/i]
1939 Moscow Mathematical Olympiad, 050
Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.
2025 Harvard-MIT Mathematics Tournament, 7
Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_1$ and $x_2,$ the number of $x_2$’s between the first and second occurrences of $x_1$ equals the number of $x_2$’s between the second and third occurrences of $x_1.$
2016 Peru IMO TST, 15
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2022 Assam Mathematical Olympiad, 2
Find the sum of all the positive divisors of $27000$.
IV Soros Olympiad 1997 - 98 (Russia), 11.7
Solve the inequality $$\log_{\frac12} x\ge 16^x$$
2006 ITAMO, 4
The squares of an infinite chessboard are numbered $1,2,\ldots $ along a spiral, as shown in the picture. A [i]rightline[/i] is the sequence of the numbers in the squares obtained by starting at one square at going to the right.
a) Prove that exists a rightline without multiples of $3$.
b) Prove that there are infinitely many pairwise disjoint rightlines not containing multiples of $3$.
2024 HMNT, 7
Let triangle $ABC$ have $AB = 5, BC = 8,$ and $\angle ABC = 60^\circ.$ A circle $\omega$ tangent to segments $AB$ and $BC$ intersects segment $CA$ at points $X$ and $Y$ such that points $C, Y , X,$ and $A$ lie along $CA$ in this order. If $\omega$ is tangent to $AB$ at point $Z$ and $ZY \parallel BC,$ compute the radius of $\omega.$
2019 Polish Junior MO Second Round, 4.
Let $ABC$ be such a triangle, that $AB = 3\cdot BC$. Points $P$ and $Q$ lies on the side $AB$ and $AP = PQ = QB$. A point $M$ is the midpoint of the side $AC$. Prove that $\sphericalangle PMQ = 90^{\circ}$.
1959 AMC 12/AHSME, 30
$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds?
$ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $
1995 National High School Mathematics League, 1
In arithmetic sequence $(a_n)$, $3a_8=5a_{13},a_1>0$. Define $S_n=\sum_{i=1}^n a_i$, then the largest number in $(S_n)$ is
$\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}$
2019 Online Math Open Problems, 24
Let $ABC$ be an acute scalene triangle with orthocenter $H$ and circumcenter $O$. Let the line through $A$ tangent to the circumcircle of triangle $AHO$ intersect the circumcircle of triangle $ABC$ at $A$ and $P \neq A$. Let the circumcircles of triangles $AOP$ and $BHP$ intersect at $P$ and $Q \neq P$. Let line $PQ$ intersect segment $BO$ at $X$. Suppose that $BX=2$, $OX=1$, and $BC=5$. Then $AB \cdot AC = \sqrt{k}+m\sqrt{n}$ for positive integers $k$, $m$, and $n$, where neither $k$ nor $n$ is divisible by the square of any integer greater than $1$. Compute $100k+10m+n$.
[i]Proposed by Luke Robitaille[/i]