Found problems: 85335
1999 IMO, 1
A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
2010 Sharygin Geometry Olympiad, 6
Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$
2019 Teodor Topan, 4
Let be an odd natural number $ n, $ and $ n $ real numbers $ y_1\le y_2\le\cdots\le y_n $ whose sum is $ 0. $ Prove that
$$ (n+2)y_{\frac{n+1}{2}}^2\le y_1^2+y_2^2+\cdots +y_n^2, $$
and specify where equality is attained.
[i]Nicolae Bourbăcuț[/i]
2012 Today's Calculation Of Integral, 814
Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.
2008 Bulgarian Autumn Math Competition, Problem 8.2
Let $\triangle ABC$ have $\angle A=20^{\circ}$ and $\angle C=40^{\circ}$. We've constructed the angle bisector $AL$ ($L\in BC$) and the external angle bisector $CN$ ($N\in AB$). Find $\angle CLN$.
2024/2025 TOURNAMENT OF TOWNS, P2
There are $100$ lines in the plane, such that no two are parallel and no three are concurrent. Consider the quadrilaterals such that all their sides lie on these lines (including the quadrilaterals whose interior is crossed by some of these lines). Is it true that the number of convex quadrilaterals equals the number of non-convex ones?
2022 China Team Selection Test, 5
Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \le q \le n-1$, such that
\[ \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, \]
where the indices are take modulo $2n$.
[i]Note:[/i] If $q=0$, then $\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$.
1993 Moldova Team Selection Test, 6
The numbers $1,2,...,2n-1,2n$ are divided into two disjoint sets, $a_1 < a_2 < ... < a_n$ and $b_1 > b_2 > ... > b_n$. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2.$$
1971 AMC 12/AHSME, 4
After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents
$\textbf{(A) }11\qquad\textbf{(B) }12\qquad\textbf{(C) }13\qquad\textbf{(D) }21\qquad \textbf{(E) }31$
2017 Harvard-MIT Mathematics Tournament, 10
Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$ such that $P_i$ beats $P_{i+1}$ for $i=1, 2, 3, 4$. (We denote $P_5=P_1$).
2000 Balkan MO, 1
Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.
2015 Iran MO (3rd round), 5
Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$
2008 Indonesia TST, 4
Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$
2009 Purple Comet Problems, 5
A train car held $6000$ pounds of mud which was $88$ percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only $82$ percent water. How many pounds does the mud weigh now?
2022 Taiwan Mathematics Olympiad, 1
Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If
\[x\mid yz(x+y+z),\]
\[y\mid xz(x+y+z),\]
\[z\mid xy(x+y+z),\]
and
\[x+y+z\mid xyz,\]
show that $xyz(x+y+z)$ is a perfect square.
[i]Proposed by usjl[/i]
2008 Sharygin Geometry Olympiad, 19
(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB \equal{} a$, $ AD \equal{} b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.
2018 Pan-African Shortlist, A5
Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying:
[list]
[*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$,
[*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and
[*] $g(x) \neq x$ for $2 \leq x \leq 2018$.
[/list]
Find the minimum possible value of $g(2)$.
2014 Balkan MO Shortlist, A1
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$
2010 ELMO Shortlist, 6
Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it.
[i]Brian Hamrick.[/i]
2009 Romanian Masters In Mathematics, 2
A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$
[i]Dan Schwarz, Romania[/i]
2012 National Olympiad First Round, 18
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$
2022 JHMT HS, 10
The maximum value of
\[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \]
over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.
1963 All Russian Mathematical Olympiad, 028
Eight men had participated in the chess tournament. (Each meets each, draws are allowed, giving $1/2$ of point, winner gets $1$.) Everyone has different number of points. The second one has got as many points as the four weakest participants together. What was the result of the play between the third prizer and the chess-player that have occupied the seventh place?
2020 Saint Petersburg Mathematical Olympiad, 2.
Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.
2017 USA Team Selection Test, 1
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.
For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.