Found problems: 85335
2010 Contests, 3
Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$.
(a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point.
(b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.
2016 EGMO TST Turkey, 6
Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying
\[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]
2011 Saudi Arabia Pre-TST, 4.1
On a semicircle of diameter $AB$ and center $C$, consider variable points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value.
2024 AMC 12/AHSME, 2
What is $10! - 7! \cdot 6!$?
$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
2019 Saint Petersburg Mathematical Olympiad, 3
Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that
$$a^b\left(a+b\right)^c>c^b a^c.$$
1995 Yugoslav Team Selection Test, Problem 2
A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.
2000 All-Russian Olympiad, 4
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$
2012 Romania Team Selection Test, 4
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
1964 Miklós Schweitzer, 3
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.
1972 IMO Shortlist, 7
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
2016 Dutch BxMO TST, 5
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$
MOAA Gunga Bowls, 2023.24
Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$.
[i]Proposed by Harry Kim[/i]
2020 AMC 10, 18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$
1997 Israel National Olympiad, 8
Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.
2006 Federal Math Competition of S&M, Problem 1
In a convex quadrilateral $ABCD$, $\angle BAC=\angle DAC=55^\circ$, $\angle DCA=20^\circ$, and $\angle BCA=15^\circ$. Find the measure of $\angle DBA$.
2011 Bulgaria National Olympiad, 1
Point $O$ is inside $\triangle ABC$. The feet of perpendicular from $O$ to $BC,CA,AB$ are $D,E,F$. Perpendiculars from $A$ and $B$ respectively to $EF$ and $FD$ meet at $P$. Let $H$ be the foot of perpendicular from $P$ to $AB$. Prove that $D,E,F,H$ are concyclic.
2020 Vietnam Team Selection Test, 2
In acute $\triangle ABC$, $O$ is the circumcenter, $I$ is the incenter. The incircle touches $BC,CA,AB$ at $D,E,F$. And the points $K,M,N$ are the midpoints of $BC,CA,AB$ respectively.
a) Prove that the lines passing through $D,E,F$ in parallel with $IK,IM,IN$ respectively are concurrent.
b) Points $T,P,Q$ are the middle points of the major arc $BC,CA,AB$ on $\odot ABC$. Prove that the lines passing through $D,E,F$ in parallel with $IT,IP,IQ$ respectively are concurrent.
2017 Purple Comet Problems, 2
The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$. Find the area of the large square.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/bad21be1b3993586c3860efa82ab27d340dbcb.png[/img]
2016 ASDAN Math Tournament, 1
Let $x$ and $y$ be positive real numbers such that $x+y=\tfrac{1}{x}+\tfrac{1}{y}=5$. Compute $x^2+y^2$.
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
2024 Baltic Way, 7
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2014 ASDAN Math Tournament, 6
In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$.
1986 All Soviet Union Mathematical Olympiad, 437
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.
2017 China Team Selection Test, 1
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.