Found problems: 85335
2006 Cuba MO, 2
Let $U$ be the center of the circle inscribed in the triangle $ABC$, $O_1$, $O_2$ and $O_3$ the centers of the circles circumscribed by the triangles $BCU$, $CAU$ and $ABU$ respectively. Prove that the circles circumscribed around the triangles $ABC$ and $O_1O_2O_3$ have the same center.
2010 QEDMO 7th, 2
Let $c: Q-\{0\} \to Q-\{0\}$ a function with the following properties (for all $x,y, a, b \in Q-\{0\}$ and $x \ne 1$):
a) $c (x, 1- x) = 1$
b) $c (ab,y) = c (a,y)c(b, y)$
c) $c (y,ab) = c (y, a)c(y,b)$
Show that then $c (a,b) c(b,a) = 1 = c(a,-a)$ also holds.
2009 Poland - Second Round, 1
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\Gamma$ with $AB$ as diameter. Let $E$ be the intersection of the diagonals $AC$ and $BD$. The tangents to $\Gamma$ at the points $C,D$ meet at $P$. Prove that $PC=PE$.
2018 Middle European Mathematical Olympiad, 4
Let $n$ be a positive integer and $u_1,u_2,\cdots ,u_n$ be positive integers not larger than $2^k, $ for some integer $k\geq 3.$ A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_1,a_2,\cdots ,a_n$ such that $t=a_1u_1+a_2u_2+\cdots +a_nu_n.$
Prove that if a non-negative integer $t$ has a representation,then it also has a representation where less than $2k$ of numbers $a_1,a_2,\cdots ,a_n$ are non-zero.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P3
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$.
[i]Proposed by Nikola Velov[/i]
2010 Belarus Team Selection Test, 2.4
Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$.
(I. Voronovich)
2008 ITest, 63
Looking for a little time alone, Michael takes a jog at along the beach. The crashing of waves reminds him of the hydroelectric plant his father helped maintain before the family moved to Jupiter Falls. Michael was in elementary school at the time. He thinks about whether he wants to study engineering in college, like both his parents did, or pursue an education in business. His aunt Jessica studied business and appraises budding technology companies for a venture capital firm. Other possibilities also tug a little at Michael for different reasons.
Michael stops and watches a group of girls who seem to be around Tony's age play a game around an ellipse drawn in the sand. There are two softball bats stuck in the sand. Michael recognizes these as the foci of the ellipse. The bats are $24$ feet apart. Two children stand on opposite ends of the ellipse where the ellipse intersects the line on which the bats lie. These two children are $40$ feet apart. Five other children stand on different points of the ellipse. One of them blows a whistle and all seven children run screaming toward one bat or the other. Each child runs as fast as she can, touching one bat, then the next, and finally returning to the spot on which she started. When the first girl gets back to her place, she declares, "I win this time! I win!" Another of the girls pats her on the back, and the winning girl speaks again. "This time I found the place where I'd have to run the shortest distance."
Michael thinks for a moment, draws some notes in the sand, then computes the shortest possible distance one of the girls could run from her starting point on the ellipse, to one of the bats, to the other bat, then back to her starting point. He smiles for a moment, then keeps jogging. If Michael's work is correct, what distance did he compute as the shortest possible distance one of the girls could run during the game?
2009 China Team Selection Test, 1
Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$
1971 Polish MO Finals, 1
Show that if $(a_n)$ is an infinite sequence of distinct positive integers, neither of which contains digit $0$ in the decimal expansion, then $$\sum_{n=1}^{\infty} \frac{1}{a_n}< 29.$$
2023 Indonesia TST, 1
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
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 $n$.
2015 BMT Spring, Tie 3
The permutohedron of order $3$ is the hexagon determined by points $(1, 2, 3)$, $(1, 3, 2)$, $(2, 1, 3)$, $(2, 3, 1)$, $(3, 1, 2)$, and $(3, 2, 1)$. The pyramid determined by these six points and the origin has a unique inscribed sphere of maximal volume. Determine its radius.
1967 AMC 12/AHSME, 39
Given the sets of consecutive integers $\{1\}$,$ \{2, 3\}$,$ \{4,5,6\}$,$ \{7,8,9,10\}$,$\; \cdots \; $, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the $N$th set. Then $S_{21}$ equals:
$\textbf{(A)}\ 1113\qquad
\textbf{(B)}\ 4641 \qquad
\textbf{(C)}\ 5082\qquad
\textbf{(D)}\ 53361\qquad
\textbf{(E)}\ \text{none of these}$
2018 Junior Balkan Team Selection Tests - Romania, 1
Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square.
Lucian Petrescu
2021 Macedonian Team Selection Test, Problem 6
Let $ABC$ be an acute triangle such that $AB<AC$ with orthocenter $H$. The altitudes $BH$ and $CH$ intersect $AC$ and $AB$ at $B_{1}$ and $C_{1}$. Denote by $M$ the midpoint of $BC$. Let $l$ be the line parallel to $BC$ passing through $A$. The circle around $ CMC_{1}$ meets the line $l$ at points $X$ and $Y$, such that $X$ is on the same side of the line $AH$ as $B$ and $Y$ is on the same side of $AH$ as $C$. The lines $MX$ and $MY$ intersect $CC_{1}$ at $U$ and $V$ respectively. Show that the circumcircles of $ MUV$ and $ B_{1}C_{1}H$ are tangent.
[i] Authored by Nikola Velov[/i]
2010 Malaysia National Olympiad, 8
For any number $x$, let $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. A sequence $a_1,a_2,\cdots$ is given, where \[a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.\]
How many values of $k$ are there such that $a_k=2010$?
2004 Dutch Mathematical Olympiad, 2
Two circles $A$ and $B$, both with radius $1$, touch each other externally.
Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that
$P$ externally touches on $A, B, Q$ and $S$,
$Q$ externally touches on $P, B$ and $R$,
$R$ externally touches on $A, B, Q$ and $S$,
$S$ externally touches on $P, A$ and $R$.
Calculate the length of $r.$
[asy]
unitsize(0.3 cm);
pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;
A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);
draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));
label("$A$", A);
label("$B$", B);
label("$P$", P);
label("$Q$", Q);
label("$R$", R);
label("$S$", S);
[/asy]
2022 Brazil Undergrad MO, 6
Let $p \equiv 3 \,(\textrm{mod}\, 4)$ be a prime and $\theta$ some angle such that $\tan(\theta)$ is rational. Prove that $\tan((p+1)\theta)$ is a rational number with numerator divisible by $p$, that is, $\tan((p+1)\theta) = \frac{u}{v}$ with $u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1$ and $u \equiv 0 \,(\textrm{mod}\,p) $.
1999 AIME Problems, 5
For any positive integer $x$, let $S(x)$ be the sum of the digits of $x$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $T(x)$ do not exceed 1999?
2022 Princeton University Math Competition, A7
Kelvin has a set of eight vertices. For each pair of distinct vertices, Kelvin independently draws an edge between them with probability $p \in (0,1).$ A set $S$ of four distinct vertices is called [i]good[/i] if there exists an edge between $v$ and $w$ for all $v,w \in S$ with $v \neq w.$ The variance of the number of good sets can be expressed as a polynomial $f(p)$ in the variable $p.$ Find the sum of the absolute values of the coefficients of $f(p).$
(The [i]variance[/i] of a random variable $X$ is defined as $\mathbb{E}[X^2]-\mathbb{E}[X]^2.$)
2013 National Olympiad First Round, 9
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$?
$
\textbf{(A)}\ 70
\qquad\textbf{(B)}\ 72
\qquad\textbf{(C)}\ 84
\qquad\textbf{(D)}\ 96
\qquad\textbf{(E)}\ 108
$
Gheorghe Țițeica 2025, P2
Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold?
[i]Dorel Miheț[/i]
2000 Harvard-MIT Mathematics Tournament, 13
Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?
1977 Spain Mathematical Olympiad, 2
Prove that all square matrices of the form (with $a, b \in R$),
$$\begin{pmatrix}
a & b \\
-b & a
\end{pmatrix}$$
form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.
2003 Austria Beginners' Competition, 2
Find all real solutions of the equation $(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3$.