Found problems: 85335
2016 Harvard-MIT Mathematics Tournament, 20
Let $ABC$ be a triangle with $AB=13$, $AC=14$, and $BC=15$. Let $G$ be the point on $AC$ such that the reflection of $BG$ over the angle bisector of $\angle B$ passes through the midpoint of $AC$. Let $Y$ be the midpoint of $GC$ and $X$ be a point on segment $AG$ such that $\frac{AX}{XG}=3$. Construct $F$ and $H$ on $AB$ and $BC$, respectively, such that $FX \parallel BG \parallel HY$. If $AH$ and $CF$ concur at $Z$ and $W$ is on $AC$ such that $WZ \parallel BG$, find $WZ$.
2016 Chile TST IMO, 3
A set \( A \) of integers is said to be \textit{admissible} if it satisfies the property:
\[
\text{If } x, y \in A, \text{ then } x^2 + kxy + y^2 \in A \text{ for all } k \in \mathbb{Z}.
\]
Determine all pairs \( (m, n) \) of nonzero integers such that the only admissible set containing both \( m \) and \( n \) is the set of all integers.
2017 Thailand Mathematical Olympiad, 7
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
.
2021 Brazil Team Selection Test, 4
Find all positive integers $n$ with the folowing property: for all triples ($a$,$b$,$c$) of positive real there is a triple of non negative integers ($l$,$j$,$k$) such that $an^k$, $bn^j$ and $cn^l$ are sides of a non degenate triangle
1951 Miklós Schweitzer, 7
Let $ f(x)$ be a polynomial with the following properties:
(i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?
2007 Bulgarian Autumn Math Competition, Problem 10.4
Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.
2023 SEEMOUS, P2
For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]
1987 Nordic, 1
Nine journalists from different countries attend a press conference. None of these speaks more than three
languages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.
1995 Polish MO Finals, 1
How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?
2020 AMC 10, 13
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$
2003 Romania Team Selection Test, 2
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.
2022 Korea Winter Program Practice Test, 3
Let $n\ge 2$ be a positive integer. $S$ is a set of $2n$ airports. For two arbitrary airports $A,B$, if there is an airway from $A$ to $B$, then there is an airway from $B$ to $A$. Suppose that $S$ has only one independent set of $n$ airports. Let the independent set $X$. Prove that there exists an airport $P\in S\setminus X$ which satisfies following condition.
[b]Condition[/b] : For two arbitrary distinct airports $A,B\in S\setminus \{P\}$, if there exists a path connecting $A$ and $B$, then there exists a path connecting $A$ and $B$ which does not pass $P$.
2002 Belarusian National Olympiad, 6
The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$
2023 Thailand October Camp, 1
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.
2018 Hong Kong TST, 1
The altitudes $AD$ and $BE$ of acute triangle $ABC$ intersect at $H$. Let $F$ be the intersection of $AB$ and a line that is parallel to the side $BC$ and goes through the circumcentre of $ABC$. Let $M$ be the midpoint of $AH$. Prove that $\angle CMF=90^\circ$
2011 AMC 10, 3
Suppose $[a \,\,\, b]$ denotes the average of $a$ and $b$, and $\{a\,\,\,b\,\,\,c\}$ denotes the average of $a$, $b$, and $c$. What is $\{\{1\,\,\, 1\,\,\, 0\}\,\,\, [0\,\,\, 1]\,\,\, 0\}$?
$ \textbf{(A)}\ \frac{2}{9} \qquad\textbf{(B)}\ \frac{5}{18} \qquad\textbf{(C)}\ \frac{1}{3} \qquad\textbf{(D)}\ \frac{7}{18} \qquad\textbf{(E)}\ \frac{2}{3} $
2012 All-Russian Olympiad, 1
Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have
\[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\]
Prove that the polynomial $P(x)$ has at least one real root.
1976 AMC 12/AHSME, 23
For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer
$\textbf{(A) }\text{for all }k\text{ and }n\qquad$
$\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$
$\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$
2023 Argentina National Olympiad, 1
Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black)
The figure can be rotated $90°, 180°$ or $270°$.
Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.
the 14th XMO, P1
Nonnegative reals $x_1$, $x_2$, $\dots$, $x_n$ satisfies $x_1+x_2+\dots+x_n=n$. Let $||x||$ be the distance from $x$ to the nearest integer of $x$ (e.g. $||3.8||=0.2$, $||4.3||=0.3$). Let $y_i = x_i ||x_i||$. Find the maximum value of $\sum_{i=1}^n y_i^2$.
PEN E Problems, 12
Show that there are infinitely many primes.
1955 Moscow Mathematical Olympiad, 290
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?
1971 IMO Shortlist, 14
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
2021 Korea National Olympiad, P1
Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.
2013 Princeton University Math Competition, 1
Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\] Find the number of possibilities for the ordered triple $(a,b,c)$.