Found problems: 85335
2019 USMCA, 15
Let $P(x)$ be a polynomial with integer coefficients such that
\[P(\sqrt{2}\sin x) = -P(\sqrt{2}\cos x)\]
for all real numbers $x$. What is the largest prime that must divide $P(2019)$?
PEN H Problems, 30
Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.
2020 Jozsef Wildt International Math Competition, W28
For positive integers $j\le n$, prove that
$$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$
[i]Proposed by Ángel Plaza[/i]
2022 Belarusian National Olympiad, 8.2
Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$.
Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square.
2013 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent.
[i]Proposed by Michael Kural[/i]
KoMaL A Problems 2022/2023, A. 844
The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$.
[i]Proposed by Márton Lovas, Budapest[/i]
2008 Sharygin Geometry Olympiad, 21
(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle.
a) All three segments are equal. Is it true that the triangle is equilateral?
b) Two segments are equal. Is it true that the triangle is isosceles?
c) Can the segments have length 4, 4 and 3?
1952 Putnam, B2
Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$
2019 Greece Junior Math Olympiad, 4
In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move:
[i]They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$.[/i]
John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?
2018 Bosnia and Herzegovina Team Selection Test, 3
Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.
2023 MIG, 6
If $a+3b = 9$ and $a+11b =21$, what is the missing coefficient in the expression $2a+\underline{?}b = 27$?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 9\qquad\textbf{(D) } 12\qquad\textbf{(E) } 14$
2019 Regional Olympiad of Mexico Center Zone, 6
Find all positive integers $m$ with the next property:
If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that
$m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$
is a perfect power.
2018 India IMO Training Camp, 3
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.
2004 Germany Team Selection Test, 1
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
2014 Benelux, 4
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.
[list]
[*] [b](a)[/b] Prove that $|BP|\ge |BR|$
[*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]
1998 AMC 8, 11
Harry has $3$ sisters and $5$ brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$?
$ \text{(A)}\ 8\qquad\text{(B)}\ 10\qquad\text{(C)}\ 12\qquad\text{(D)}\ 15\qquad\text{(E)}\ 18 $
2023 Balkan MO Shortlist, A2
Let $a, b, c, d$ be non-negative reals such that $\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1$. Show that there exists a permutation $(x_1, x_2, x_3, x_4)$ of $(a, b, c, d)$, such that $$x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.$$
2018 AMC 12/AHSME, 6
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2022 AIME Problems, 12
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that $$\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.$$ Find the least possible value of $a+b.$
MOAA Team Rounds, 2021.5
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
[asy]
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
[/asy]
[i]Proposed by Nathan Xiong[/i]
1988 Tournament Of Towns, (182) 5
A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick .
(A . Andjans , Riga)
PEN S Problems, 38
The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.
2019 Hanoi Open Mathematics Competitions, 1
Let $x$ and $y$ be positive real numbers. Which of the following expressions is larger than both $x$ and $y$?
[b]A.[/b] $xy + 1$ [b]B.[/b] $(x + y)^2$ [b]C.[/b] $x^2 + y$ [b]D.[/b] $x(x + y)$ [b]E.[/b] $(x + y + 1)^2$
MIPT student olimpiad spring 2023, 1
In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers.
Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched
by these vectors, is the product of an integer and $\sqrt(n)$.
2024 LMT Fall, 16
Let $ZHAO$ be a square with area $2024$. Let $X$ be the center of this square and let $C$, $D$, $E$, $K$ be the centroids of $XZH$, $XHA$, $XAO$, and $XOZ$, respectively. Find $[ZHAO]$ $+$ $[CZHAO]$ $+$ $[DZHAO]$ $+$ $[EZHAO]$ $+$ $[KZHAO]$.
(Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)