Found problems: 85335
2023 South Africa National Olympiad, 6
Let $ABIH$,$BDEC$ and $ACFG$ be arbitrary rectangles constructed (externally) on the sides of triangle $ABC$.Choose point $S$ outside rectangle $ABIH$ (on the opposite side as triangle $ABC$) such that $\angle SHI=\angle FAC$ and $\angle HIS=\angle EBC$.Prove that the lines $FI,EH$ and $CS$ are concurrent(i.e., the three lines intersect in one point).
2010 Romania National Olympiad, 1
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
2013 Online Math Open Problems, 11
Four orange lights are located at the points $(2,0)$, $(4,0)$, $(6,0)$ and $(8,0)$ in the $xy$-plane. Four yellow lights are located at the points $(1,0)$, $(3,0)$, $(5,0)$, $(7,0)$. Sparky chooses one or more of the lights to turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric around some line parallel to the $y$-axis?
[i]Proposed by Evan Chen[/i]
1995 Nordic, 4
Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying
(i) The side lengths of $T $ are consecutive integers.
(ii) The area of $T$ is an integer.
2006 AMC 10, 14
Let $ a$ and $ b$ be the roots of the equation $ x^2 \minus{} mx \plus{} 2 \equal{} 0$. Suppose that $ a \plus{} (1/b)$ and $ b \plus{} (1/a)$ are the roots of the equation $ x^2 \minus{} px \plus{} q \equal{} 0$. What is $ q$?
$ \textbf{(A) } \frac 52 \qquad \textbf{(B) } \frac 72 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } \frac 92 \qquad \textbf{(E) } 8$
2023 China Team Selection Test, P18
Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$.
Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.
2000 Croatia National Olympiad, Problem 2
Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.
2010 USAMO, 6
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
2008 AMC 8, 14
Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?
[asy]
size((80));
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
draw((0,3)--(9,3));
draw((0,6)--(9,6));
label("A", (1.5,7.5));
[/asy]
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $
2021 China Team Selection Test, 5
Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.
2013 Tournament of Towns, 5
Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).
2006 Iran MO (3rd Round), 3
Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]
2015 Macedonia National Olympiad, Problem 3
All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.
2007 Today's Calculation Of Integral, 233
Find the minimum value of the following definite integral.
$ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$
2010 N.N. Mihăileanu Individual, 3
Consider a countinuous function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that verifies the following conditions:
$ \text{(1)} x f(f(x))=(f(x))^2,\quad\forall x\in\mathbb{R}_{>0} $
$ \text{(2)} \lim_{\stackrel{x\to 0}{x>0}} \frac{f(x)}{x}\in\mathbb{R}\cup\{ \pm\infty \} $
[b]a)[/b] Show that $ f $ is bijective.
[b]b)[/b] Prove that the sequences $ \left( (\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}} ) (x) \right)_{n\ge 1} ,\left( (\underbrace{f^{-1}\circ f^{-1}\circ\cdots \circ f^{-1}}_{\text{n times}} ) (x) \right)_{n\ge 1} $ are both arithmetic progressions, for any fixed $ x\in\mathbb{R}_{>0} . $
[b]c)[/b] Determine the function $ f. $
[i]Nelu Chichirim[/i]
1997 Italy TST, 2
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
2017 AMC 10, 11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2017 Korea National Olympiad, problem 2
Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following.
$$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$
1949 Moscow Mathematical Olympiad, 164
There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?
MIPT student olimpiad spring 2024, 1
Find integral:
$\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$
1965 AMC 12/AHSME, 28
An escalator (moving staircase) of $ n$ uniform steps visible at all times descends at constant speed. Two boys, $ A$ and $ Z$, walk down the escalator steadily as it moves, $ A$ negotiating twice as many escalator steps per minute as $ Z$. $ A$ reaches the bottom after taking $ 27$ steps while $ Z$ reaches the bottom after taking $ 18$ steps. Then $ n$ is:
$ \textbf{(A)}\ 63 \qquad \textbf{(B)}\ 54 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 30$
1998 Belarus Team Selection Test, 3
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.
1949-56 Chisinau City MO, 51
Determine graphically the number of roots of the equation $\sin x = \lg x$.
2022-2023 OMMC, 7
Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$
2023 Bulgaria JBMO TST, 4
Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).