Found problems: 85335
2006 Sharygin Geometry Olympiad, 21
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$
and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.
Kvant 2023, M2739
In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.
2019 CCA Math Bonanza, T3
What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$?
[i]2019 CCA Math Bonanza Team Round #3[/i]
2010 AIME Problems, 7
Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$. For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $ \{1,2,3,4,5,6,7\}$. Find the remainder when $ N$ is divided by $ 1000$.
[b]Note[/b]: $ |S|$ represents the number of elements in the set $ S$.
2019 APMO, 5
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$.
1994 All-Russian Olympiad, 3
There are three piles of matches on the table: one with $100$ matches, one with $200$, and one with $300$. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy?
(K. Kokhasā)
1999 All-Russian Olympiad Regional Round, 11.6
The cells of a $50\times 50$ square are painted in four colors. Prove that there is a cell, on four sides of which (i.e. top, bottom, left and on the right) there are cells of the same color as it (not necessarily adjacent to this cell).
2010 All-Russian Olympiad Regional Round, 11.5
The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trĪ±iangle is acute-angled.
2005 AIME Problems, 7
Let \[x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.\] Find $(x+1)^{48}$.
2017 ITAMO, 4
Let $ABCD$ be a thetraedron with the following propriety: the four lines connecting a vertex and the incenter of opposite face are concurrent. Prove $AB \cdot CD= AC \cdot BD = AD\cdot BC$.
2014 Chile TST Ibero, 2
Let $\triangle ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC,$ and $CA$ respectively, such that:
\[
\frac{AP}{AB} = \frac{BQ}{BC} = \frac{CR}{CA} = \frac{1}{n}
\]
for $n \in \mathbb{N}$. The segments $AQ$ and $CP$ intersect at $D$, the segments $BR$ and $AQ$ intersect at $E$, and the segments $BR$ and $CP$ intersect at $F$. Compute the ratio:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}.
\]
1998 IMC, 3
Let $f(x)=2x(1-x), x\in\mathbb{R}$ and denote $f_n=f\circ f\circ ... \circ f$, $n$ times.
(a) Find $\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx$.
(b) Now compute $\int^1_0 f_n(x)dx$.
1990 Tournament Of Towns, (253) 1
Construct a triangle given two of its side lengths if it is known that the median drawn from their common vertex divides the angle between them in the ratio $1:2$.
(V. Chikin)
JBMO Geometry Collection, 2017
Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.
2009 National Olympiad First Round, 29
$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC \plus{} BD \equal{} 18$, then area of $ ABCD$ is ?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ \frac {81}{2} \qquad\textbf{(C)}\ \frac {36\sqrt 3}{2} \qquad\textbf{(D)}\ \frac {81\sqrt 3}{4} \qquad\textbf{(E)}\ \text{None}$
2022 Pan-African, 3
Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define
$$
A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.
$$
Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.
2019 Bundeswettbewerb Mathematik, 4
Prove that for no integer $k \ge 2$, between $10k$ and $10k + 100$ there are more than $23$ prime numbers.
2010 Contests, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
1996 All-Russian Olympiad Regional Round, 11.8
Is there an infinite periodic sequence consisting of the letters $a$ and$ b$, such that if all letters are replaced simultaneously $a$ to $aba$ and letters $b$ to $bba$ does it transform into itself (possibly with a shift)? (A sequence is called periodic if there is such natural number $n$, which for every $i = 1, 2, . . . i$-th member of this sequence is equal to the ($i + n$)- th.)
1997 IMO Shortlist, 13
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$
II Soros Olympiad 1995 - 96 (Russia), 9.1
Solve the inequality
$$(x-1)(x^2-1)(x^3-1)\cdot ...\cdot (x^{100}-1)(x^{101}-1)\ge 0$$
2018 Junior Regional Olympiad - FBH, 1
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?
2010 Rioplatense Mathematical Olympiad, Level 3, 3
Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy
\[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$.
Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.
1991 AMC 12/AHSME, 15
A circular table has exactly 60 chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of $N$ is
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 58 $
1997 AMC 8, 16
Penni Precisely buys $\$100$ worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up $20\%$, BB was down $25\%$, and CC was unchanged. For the second year, AA was down $20\%$ from the previous year, BB was up $25\%$ from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then
$\textbf{(A)}\ A=B=C \qquad \textbf{(B)}\ A=B<C \qquad \textbf{(C)}\ C<B=A$
$\textbf{(D)}\ A<B<C \qquad \textbf{(E)}\ B<A<C$