Found problems: 85335
2008 All-Russian Olympiad, 3
A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.
2018 OMMock - Mexico National Olympiad Mock Exam, 3
Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds:
\[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \]
[i]Proposed by Oriol Solé[/i]
2008 Peru Iberoamerican Team Selection Test, P1
For every integer $m>1$, let $p(m)$ be the least prime divisor of $m$. If $a$ and $b$ are integers greater than $1$ such that:
$$a^2+b=p(a)+[p(b)]^2$$
Show that $a=b$
2007 iTest Tournament of Champions, 1
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$.
1979 VTRMC, 7
Let S be a finite set of non-negative integers such that $| x - y | \in S$ whenever $x , y \in S$.
(a) Give an example of such a set which contains ten elements.
(b) If $A$ is a subset of $S$ containing more than two-thirds of the elements of $S$, prove or disprove that [i]every[/i] element of $S$ is the sum or difference of two elements from $A$.
2024 European Mathematical Cup, 3
Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$
1951 Moscow Mathematical Olympiad, 206
Consider a curve with the following property:
[i]inside the curve one can move an inscribed equilateral triangle so that each vertex of the triangle moves along the curve and draws the whole curve[/i].
Clearly, every circle possesses the property. Find a closed planar curve without self-intersections, that has the property but is not a circle.
2015 Thailand Mathematical Olympiad, 2
Let $a, b, c$ be positive reals with $abc = 1$. Prove the inequality
$$\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32$$
and determine all values of a, b, c for which equality is attained
2009 IMO Shortlist, 2
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2021 Argentina National Olympiad, 2
In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$
1986 IMO Longlists, 54
Find the least integer $n$ with the following property:
For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points.
2006 China Second Round Olympiad, 2
Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula
\[ a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},\]
Find $a_n$.
2005 National High School Mathematics League, 6
Set $T=\{0,1,2,3,4,5,6\},M=\left\{\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4}|a_i\in T,i=1,2,3,4\right\}$. Put all elements in $M$ in order: from small to large, then the 2005th number is
$\text{(A)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{3}{7^4}$
$\text{(B)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{2}{7^4}$
$\text{(C)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{4}{7^4}$
$\text{(D)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{3}{7^4}$
2012 AMC 12/AHSME, 21
Let $a,b,$ and $c$ be positive integers with $a\ge b\ge c$ such that
\begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\
a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*}
What is $a$?
$ \textbf{(A)}\ 249
\qquad\textbf{(B)}\ 250
\qquad\textbf{(C)}\ 251
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ 253
$
2014 Contests, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
2002 Putnam, 1
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?
2022 Saudi Arabia BMO + EGMO TST, 1.3
Given is triangle $ABC$ with $AB > AC$. Circles $O_B$, $O_C$ are inscribed in angle $BAC$ with $O_B$ tangent to $AB$ at $B$ and $O_C$ tangent to $AC$ at $C$. Tangent to $O_B$ from $C$ different than $AC$ intersects $AB$ at $K$ and tangent to $O_C$ from $B$ different than $AB$ intersects $AC$ at $L$. Line $KL$ and the angle bisector of $BAC$ intersect $BC$ at points $P$ and $M$, respectively. Prove that $BP = CM$.
2010 Estonia Team Selection Test, 1
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$,
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
1996 USAMO, 1
Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.
1956 AMC 12/AHSME, 46
For the equation $ \frac {1 \plus{} x}{1 \minus{} x} \equal{} \frac {N \plus{} 1}{N}$ to be true where $ N$ is positive, $ x$ can have:
$ \textbf{(A)}\ \text{any positive value less than }1 \qquad\textbf{(B)}\ \text{any value less than }1$
$ \textbf{(C)}\ \text{the value zero only} \qquad\textbf{(D)}\ \text{any non \minus{} negative value} \qquad\textbf{(E)}\ \text{any value}$
2017 Saint Petersburg Mathematical Olympiad, 4
Each cell of a $3\times n$ table was filled by a number. In each of three rows, the number $1,2,…,n$ appear in some order. It is know that for each column, the sum of pairwise product of three numbers in it is a multiple of $n$. Find all possible value of $n$.
2006 China National Olympiad, 6
Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.
2015 İberoAmerican, 6
Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$, in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns.
2019 VJIMC, 3
For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$ ,$M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$.
Find the smallest integer $n\geq 2 $ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$.
($M^T$ means the transpose of a matrix $M$ . A square matrix is called normal if $M^T M=M M^T$ holds).
[i]Proposed by Martin Niepel (Comenius University, Bratislava)..[/i]
2014 Argentine National Olympiad, Level 3, 3.
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.