Found problems: 85335
1997 Croatia National Olympiad, Problem 1
In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that:
(a) $ABL$ and $DMLN$ have equal areas;
(b) $\angle ALD=\angle OLN=60^\circ$;
(c) $\angle OLD=90^\circ$.
2018 AIME Problems, 5
Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.
2013 India IMO Training Camp, 3
In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.
2019 BMT Spring, 8
For a positive integer $ n $, define $ \phi(n) $ as the number of positive integers less than or equal to $ n $ that are relatively prime to $ n $. Find the sum of all positive integers $ n $ such that $ \phi(n) = 20 $.
2008 Junior Balkan Team Selection Tests - Moldova, 4
The square table $ 10\times 10$ is divided in squares $ 1\times1$. In each square $ 1\times1$ is written one of the numers $ \{1,2,3,...,9,10\}$. Numbers from any two adjacent or diagonally adjacent squares are reciprocal prime. Prove, that there exists a number, which is written in this table at least 17 times.
1987 AIME Problems, 10
Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)
2023 AMC 8, 2
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
[asy]
//kante314
size(11cm);
filldraw((0,0)--(29,0)--(29,29)--(0,29)--cycle,mediumgray);
draw((36,29/2)--(54,29/2),EndArrow(size=7));
draw((36,29/2)--(52.5,29/2),linewidth(1.5));
filldraw((61,22)--(63,22)--(63,6)--cycle,mediumgray);
fill((63,6+1*17/16)--(80,6+1*17/16)--(80,6+2*17/16)--(63,6+2*17/16)--cycle,lightgray);
fill((63,6+3*17/16)--(80,6+3*17/16)--(80,6+4*17/16)--(63,6+4*17/16)--cycle,lightgray);
fill((63,6+5*17/16)--(80,6+5*17/16)--(80,6+6*17/16)--(63,6+6*17/16)--cycle,lightgray);
fill((63,6+7*17/16)--(80,6+7*17/16)--(80,6+8*17/16)--(63,6+8*17/16)--cycle,lightgray);
fill((63,6+9*17/16)--(80,6+9*17/16)--(80,6+10*17/16)--(63,6+10*17/16)--cycle,lightgray);
fill((63,6+11*17/16)--(80,6+11*17/16)--(80,6+12*17/16)--(63,6+12*17/16)--cycle,lightgray);
fill((63,6+13*17/16)--(80,6+13*17/16)--(80,6+14*17/16)--(63,6+14*17/16)--cycle,lightgray);
fill((63,6+15*17/16)--(80,6+15*17/16)--(80,6+16*17/16)--(63,6+16*17/16)--cycle,lightgray);
draw((63,6)--(63,23)--(68,23)--(69,12)--(80,6)--cycle);
filldraw((69,12)--(69,27)--(67,28)--cycle,mediumgray);
filldraw((69,12)--(69,29)--(80,23)--(80,6)--cycle,white);
fill((69,12+1*15/13)--(80,6+1*15/13)--(80,6+2*15/13)--(69,12+2*15/13)--cycle,lightgray);
fill((69,12+3*15/13)--(80,6+3*15/13)--(80,6+4*15/13)--(69,12+4*15/13)--cycle,lightgray);
fill((69,12+5*15/13)--(80,6+5*15/13)--(80,6+6*15/13)--(69,12+6*15/13)--cycle,lightgray);
fill((69,12+7*15/13)--(80,6+7*15/13)--(80,6+8*15/13)--(69,12+8*15/13)--cycle,lightgray);
fill((69,12+9*15/13)--(80,6+9*15/13)--(80,6+10*15/13)--(69,12+10*15/13)--cycle,lightgray);
fill((69,12+11*15/13)--(80,6+11*15/13)--(80,6+12*15/13)--(69,12+12*15/13)--cycle,lightgray);
fill((69,12+13*15/13)--(80,6+13*15/13)--(80,6+14*15/13)--(69,12+14*15/13)--cycle,lightgray);
draw((69,12)--(69,29)--(80,23)--(80,6)--cycle);
draw((87,29/2)--(105,29/2),EndArrow(size=7));
draw((87,29/2)--(102.5,29/2),linewidth(1.5));
fill((112,6+1*17/16)--(129,6+1*17/16)--(129,6+2*17/16)--(112,6+2*17/16)--cycle,lightgray);
fill((112,6+3*17/16)--(129,6+3*17/16)--(129,6+4*17/16)--(112,6+4*17/16)--cycle,lightgray);
fill((112,6+5*17/16)--(129,6+5*17/16)--(129,6+6*17/16)--(112,6+6*17/16)--cycle,lightgray);
fill((112,6+7*17/16)--(129,6+7*17/16)--(129,6+8*17/16)--(112,6+8*17/16)--cycle,lightgray);
fill((112,6+9*17/16)--(129,6+9*17/16)--(129,6+10*17/16)--(112,6+10*17/16)--cycle,lightgray);
fill((112,6+11*17/16)--(129,6+11*17/16)--(129,6+12*17/16)--(112,6+12*17/16)--cycle,lightgray);
fill((112,6+13*17/16)--(129,6+13*17/16)--(129,6+14*17/16)--(112,6+14*17/16)--cycle,lightgray);
fill((112,6+15*17/16)--(129,6+15*17/16)--(129,6+16*17/16)--(112,6+16*17/16)--cycle,lightgray);
draw((112,6)--(129,6)--(129,23)--(112,23)--cycle);
draw((112+17/2,6)--(129,6+17/2),dashed+linewidth(.3));
draw((111.7,6.7)--(111.7,23.3)--(128.3,23.3),linewidth(1));
draw((111.75,6.6)--(111.75,6.3));
draw((128.4,23.25)--(128.7,23.25));
[/asy]
[asy]
//kante314
size(11cm);
label(scale(.85)*"\textbf{(A)}", (2,55));
filldraw((7,31)--(13,31)--(19.5,37)--(26,31)--(32,31)--(32,37)--(26,43.5)--(32,50)--(32,56)--(26,56)--(19.5,50)--(13,56)--(7,56)--(7,50)--(13,43.5)--(7,37)--cycle,mediumgray);
label(scale(.85)*"\textbf{(B)}", (44,55));
filldraw((49,31)--(55,31)--(61.5,37)--(68,31)--(74,31)--(74,37)--(74,50)--(74,56)--(68,56)--(61.5,50)--(55,56)--(49,56)--(49,50)--(49,37)--cycle,mediumgray);
label(scale(.85)*"\textbf{(C)}", (86,55));
filldraw((91,31)--(116,31)--(116,56)--(91,56)--cycle,mediumgray);
filldraw((91+25/4,31+25/4)--(116-25/4,31+25/4)--(116-25/4,56-25/4)--(91+25/4,56-25/4)--cycle,white);
label(scale(.85)*"\textbf{(D)}", (2,24));
filldraw((7,0)--(32,0)--(32,25)--(7,25)--cycle,mediumgray);
filldraw((7+25/4,25/2)--(32-25/4,25/2)--(7+25/2,25-25/4)--cycle,white);
label(scale(.85)*"\textbf{(E)}", (44,24));
filldraw((49,0)--(74,0)--(74,25)--(49,25)--cycle,mediumgray);
filldraw((49+25/4,25/2)--(49+25/2,25/4)--(74-25/4,25/2)--(49+25/2,25-25/4)--cycle,white);
[/asy]
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
2021 AIME Problems, 6
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
LMT Guts Rounds, 6
Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?
2014 Dutch BxMO/EGMO TST, 4
Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.
2009 National Olympiad First Round, 3
If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ?
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$
1953 Moscow Mathematical Olympiad, 248
a) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}$
b) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\
... \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$
1979 IMO Longlists, 68
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
2003 Croatia National Olympiad, Problem 1
Show that a triangle whose side lengths are prime numbers cannot have integer area.
2009 AIME Problems, 12
In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2004 India IMO Training Camp, 2
Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$
2020 South East Mathematical Olympiad, 6
In a quadrilateral $ABCD$, $\angle ABC=\angle ADC <90^{\circ}$. The circle with diameter $AC$ intersects $BC$ and $CD$ again at $E,F$, respectively. $M$ is the midpoint of $BD$, and $AN \perp BD$ at $N$.
Prove that $M,N,E,F$ is concyclic.
2019 Kazakhstan National Olympiad, 2
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$. It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
2023 Bulgarian Spring Mathematical Competition, 11.4
Given is a tree $G$ with $2023$ vertices. The longest path in the graph has length $2n$. A vertex is called good if it has degree at most $6$. Find the smallest possible value of $n$ if there doesn't exist a vertex having $6$ good neighbors.
2006 Estonia Team Selection Test, 4
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
2019 IFYM, Sozopol, 5
Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation
$ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.
2016 India PRMO, 2
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
1995 All-Russian Olympiad Regional Round, 10.1
Given function $f(x) = \dfrac{1}{\sqrt[3]{1-x^3}}$, find $\underbrace{f(... f(f(19))...)}_{95}$.
.
2016 Harvard-MIT Mathematics Tournament, 12
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges.
[asy]
size(3cm);
draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1));
[/asy]
Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)