Found problems: 85335
2017 Tournament Of Towns, 1
A chess tournament had 10 participants. Each round, the participants split into pairs, and
each pair played a game. In total, each participant played with every other participant
exactly once, and in at least half of the games both the players were from the same town.
Prove that during each round there was a game played by two participants from the same
town.
[i](Boris Frenkin)[/i]
2009 Irish Math Olympiad, 5
Hello.
Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$.
Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality.
1960 Putnam, B6
Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that
$$\sum_{n=1}^{\infty} b_n$$
converges.
1993 Baltic Way, 20
Let $ \mathcal Q$ be a unit cube. We say that a tetrahedron is [b]good[/b] if all its edges are equal and all of its vertices lie on the boundary of $ \mathcal Q$. Find all possible volumes of good tetrahedra.
2022 Princeton University Math Competition, A1 / B3
Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt3$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ forrelatively prime positive integers $m, n$, find $m + n$.
(Two circles intersect orthogonally at a point $P$ if the tangent lines at $P$ form a right angle.)
1986 China Team Selection Test, 2
Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that:
i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}.
ii) The same as above replacing "area" for "perimeter".
2019 BMT Spring, 10
A [i]3-4-5 point[/i] of a triangle $ ABC $ is a point $ P $ such that the ratio $ AP : BP : CP $ is equivalent to the ratio $ 3 : 4 : 5 $. If $ \triangle ABC $ is isosceles with base $ BC = 12 $ and $ \triangle ABC $ has exactly one $ 3-4-5 $ point, compute the area of $ \triangle ABC $.
2022 MOAA, 2
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.
2017 Purple Comet Problems, 23
The familiar $3$-dimensional cube has $6$ $2$-dimensional faces, $12$ $1$-dimensional edges, and $8$ $0$-dimensional vertices. Find the number of $9$-dimensional sub-subfaces in a $12$-dimensional cube.
2024 Malaysian IMO Training Camp, 3
Find all primes $p$ such that for any integer $k$, there exist two integers $x$ and $y$ such that $$x^3+2023xy+y^3 \equiv k \pmod p$$
[i]Proposed by Tristan Chaang Tze Shen[/i]
2024 Indonesia TST, N
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]
2023 Thailand Mathematical Olympiad, 7
Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area
2012 Iran MO (3rd Round), 1
Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.
2017 Harvard-MIT Mathematics Tournament, 4
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.
1995 Tournament Of Towns, (443) 3
Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$.
(S Markelov)
2018 Tuymaada Olympiad, 3
$n$ rooks and $k$ pawns are arranged on a $100 \times 100$ board. The rooks cannot leap over pawns. For which minimum $k$ is it possible that no rook can capture any other rook?
Junior League: $n=2551$ ([i]Proposed by A. Kuznetsov[/i])
Senior League: $n=2550$ ([i]Proposed by N. Vlasova[/i])
1953 Miklós Schweitzer, 5
Show that any positive integer has at least as many positive divisors of the form $3k+1$ as of the form $3k-1$. [b](N. 7)[/b]
2017 Taiwan TST Round 2, 2
Find all tuples of positive integers $(a,b,c)$ such that
$$a^b+b^c+c^a=a^c+b^a+c^b$$
1955 Kurschak Competition, 2
How many five digit numbers are divisible by $3$ and contain the digit $6$?
1983 Putnam, B2
For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are:
$$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.
1967 German National Olympiad, 4
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
2022 Junior Balkan Mathematical Olympiad, 2
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
2004 India IMO Training Camp, 2
Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule:
(a) $g$ is nondecrasing
(b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$,
Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$
2014 Junior Balkan Team Selection Tests - Moldova, 7
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.