Found problems: 85335
2012 Polish MO Finals, 4
$n$ players ($n \ge 4$) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players $(A, B, C, D)$, such that $A$ won with $B$, $B$ won with $C$, $C$ won with $D$ and $D$ won with $A$. Determine, depending on $n$, maximum number of trios of players $(A, B, C)$, such that $A$ won with $B$, $B$ won with $C$ and $C$ won with $A$.
(Attention: Trios $(A, B, C)$, $(B, C, A)$ and $(C, A, B)$ are the same trio.)
2024 USEMO, 5
Let $ABC$ be a scalene triangle whose incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Lines $BE$ and $CF$ meet at $G$. Prove that there exists a point $X$ on the circumcircle of triangle $EFG$ such that the circumcircles of triangles $BCX$ and $EFG$ are tangent, and \[\angle BGC = \angle BXC + \angle EDF.\]
[i]Kornpholkrit Weraarchakul[/i]
2013 Sharygin Geometry Olympiad, 17
An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.
1967 Poland - Second Round, 4
Solve the equation in natural numbers $$
xy+yz+zx = xyz + 2.
$$
2013 Abels Math Contest (Norwegian MO) Final, 4a
An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called [i]crossing [/i] if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many [i]crossing [/i] quadruples are there in the $2013$-gon?
2013 AMC 10, 1
A taxi ride costs $\$1.50$ plus $\$0.25$ per mile traveled. How much does a $5$-mile taxi ride cost?
$ \textbf{(A)}\ \$2.25 \qquad\textbf{(B)}\ \$2.50 \qquad\textbf{(C)}\ \$2.75 \qquad\textbf{(D)}\ \$3.00 \qquad\textbf{(E)}\ \$ 3.25$
2017 Czech-Polish-Slovak Junior Match, 6
On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.
2012 District Olympiad, 1
Let $a$ and $b$ be distinct positive real numbers, such that $a -\sqrt{ab}$ and $b -\sqrt{ab}$ are both rational numbers. Prove that $a$ and $b$ are rational numbers.
2019 IFYM, Sozopol, 8
Solve the following equation in integers:
$4n^4+7n^2+3n+6=m^3$.
1987 Romania Team Selection Test, 6
The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons.
[i]Gabriel Nagy[/i]
2024 JHMT HS, 10
One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.
2016 China Western Mathematical Olympiad, 4
For an $n$-tuple of integers, define a transformation to be:
$$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$
Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$, such that for any $n$-tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$, after a finite number of transformations, every element in the of the $n$-tuple is a multiple of $k$.
2017 Swedish Mathematical Competition, 6
Let $a,b,c,x,y,z$ be real numbers such that $x+y+z=0$, $a+b+c\geq 0$, $ab+bc+ca \ge 0$. Prove that
$$ ax^2+by^2+cz^2\ge 0 $$
2014 Harvard-MIT Mathematics Tournament, 6
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.
1982 IMO Longlists, 36
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.
1990 All Soviet Union Mathematical Olympiad, 518
An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.
1997 AMC 8, 10
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.
[asy]
unitsize(8);
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);
draw((0,6)--(0,0)--(6,0));
[/asy]
$\textbf{(A)}\ \dfrac{5}{12} \qquad \textbf{(B)}\ \dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{7}{12} \qquad \textbf{(D)}\ \dfrac{2}{3} \qquad \textbf{(E)}\ \dfrac{5}{6}$
2023 Bangladesh Mathematical Olympiad, P9
Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.
2018 MIG, 12
A unit cube is sliced by a plane passing through two of its vertices and the midpoints of the edges it passes through. What is the area of the rhombus formed by this intersection?
[center][img]https://cdn.artofproblemsolving.com/attachments/3/5/3ed19fa0b4d454a3afc16c6bcf9d69403f6b2c.png[/img][/center]
$\textbf{(A) } \dfrac{\sqrt6}{2}\qquad\textbf{(B) }\sqrt2\qquad\textbf{(C) }\sqrt3\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }2\sqrt6$
2020 Final Mathematical Cup, 4
Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.
2001 Estonia Team Selection Test, 4
Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.
Durer Math Competition CD Finals - geometry, 2015.D1
From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.
2024 Harvard-MIT Mathematics Tournament, 5
The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.
2024 HMNT, 10
For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$
2021 Chile National Olympiad, 4
Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$
[img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]