Found problems: 85335
2015 AoPS Mathematical Olympiad, 7
Let $ABC$ be a right triangle with $\angle C = 90^\circ$. Let $P_A$, $P_B$, and $P_C$ be regular pentagons with side lengths $BC$, $CA$, and $AB$, respectively. Prove that $[P_A]+[P_B]=[P_C]$.
[i]Proposed by CaptainFlint[/i]
1999 German National Olympiad, 6a
Suppose that an isosceles right-angled triangle is divided into $m$ acute-angled triangles. Find the smallest possible $m$ for which this is possible.
1951 Moscow Mathematical Olympiad, 203
A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then
(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and
(2) the vertices of the faces would lie on a circle centered at $H$.
2007 Thailand Mathematical Olympiad, 11
Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.
2024 Harvard-MIT Mathematics Tournament, 9
Let $ABC$ be a triangle. Let $X$ be the point on side $AB$ such that $\angle{BXC} = 60^{\circ}$. Let $P$ be the point on segment $CX$ such that $BP\bot AC$. Given that $AB = 6, AC = 7,$ and $BP = 4,$ compute $CP$.
2023 Mid-Michigan MO, 5-6
[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$
($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits).
[b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table:
the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game.
a) The player who makes the first move can win the game. What is the winning first move?
b) How can he win? (Describe his strategy.)
[b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover?
[b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back.
[b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line.
(b) Do the same with $6$ points.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Estonia Open Junior - geometry, 2018.1.5
Let $M$ be the intersection of the diagonals of a cyclic quadrilateral $ABCD$. Find the length of $AD$, if it is known that $AB=2$ mm , $BC = 5$ mm, $AM = 4$ mm, and $\frac{CD}{CM}= 0.6$.
2019 Nigerian Senior MO Round 4, 4
We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
2017 Kosovo National Mathematical Olympiad, 4
Prove the identity
$\sum_{k=2}^{n} k(k-1)\binom{n}{k} =\binom{n}{2} 2^{n-1}$ for all $n=2,3,4,...$
2012 China National Olympiad, 1
In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$.
[hide="Diagram"][asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(14.4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */
/* draw figures */
draw(circle((-1.32,1.36), 2.98));
draw(circle((3.56,1.53), 3.18));
draw((0.92,3.31)--(-2.72,-1.27));
draw(circle((0.08,0.25), 3.18));
draw((-2.72,-1.27)--(3.13,-0.65));
draw((3.13,-0.65)--(0.92,3.31));
draw((0.92,3.31)--(2.71,-1.54));
draw((-2.41,-1.74)--(0.92,3.31));
draw((0.92,3.31)--(1.05,-0.43));
/* dots and labels */
dot((-1.32,1.36),dotstyle);
dot((0.92,3.31),dotstyle);
label("$A$", (0.81,3.72), NE * labelscalefactor);
label("$c_1$", (-2.81,3.53), NE * labelscalefactor);
dot((3.56,1.53),dotstyle);
label("$c_2$", (3.43,3.98), NE * labelscalefactor);
dot((1.05,-0.43),dotstyle);
label("$P$", (0.5,-0.43), NE * labelscalefactor);
dot((-2.72,-1.27),dotstyle);
label("$B$", (-3.02,-1.57), NE * labelscalefactor);
dot((2.71,-1.54),dotstyle);
label("$E$", (2.71,-1.86), NE * labelscalefactor);
dot((3.13,-0.65),dotstyle);
label("$C$", (3.39,-0.9), NE * labelscalefactor);
dot((-2.41,-1.74),dotstyle);
label("$D$", (-2.78,-2.07), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy][/hide]
2008 Tournament Of Towns, 2
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?
1994 AMC 8, 22
The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is
[asy]
draw(circle((0,0),3));
draw(circle((7,0),3));
draw((0,0)--(3,0));
draw((0,-3)--(0,3));
draw((7,3)--(7,0)--(7+3*sqrt(3)/2,-3/2));
draw((7,0)--(7-3*sqrt(3)/2,-3/2));
draw((0,5)--(0,3.5)--(-0.5,4));
draw((0,3.5)--(0.5,4));
draw((7,5)--(7,3.5)--(6.5,4));
draw((7,3.5)--(7.5,4));
label("$3$",(-0.75,0),W);
label("$1$",(0.75,0.75),NE);
label("$2$",(0.75,-0.75),SE);
label("$6$",(6,0.5),NNW);
label("$5$",(7,-1),S);
label("$4$",(8,0.5),NNE);
[/asy]
$\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{4}{9}$
2010 Denmark MO - Mohr Contest, 3
Can $29$ boys and $31$ girls be lined up holding hands so no one is holding hands with two girls?
2015 South East Mathematical Olympiad, 6
Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.
Ukrainian TYM Qualifying - geometry, I.17
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$.
2010 Germany Team Selection Test, 3
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
II Soros Olympiad 1995 - 96 (Russia), 11.6
The bases of the trapezoid are equal to $a$ and $b$. It is known that through the midpoint of one of its sides it is possible to draw a straight line dividing the trapezoid into two quadrangles, into each of which a circle can be inscribed. Find the length of the other side of this trapezoid.
2025 Harvard-MIT Mathematics Tournament, 3
Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.
2007 Harvard-MIT Mathematics Tournament, 2
A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors?
2023 USEMO, 1
A positive integer $n$ is called [i]beautiful[/i] if, for every integer $4 \le b \le 10000$, the base-$b$ representation of $n$ contains the consecutive digits $2$, $0$, $2$, $3$ (in this order, from left to right). Determine whether the set of all beautiful integers is finite.
[i]Oleg Kryzhanovsky[/i]
1988 IMO Longlists, 54
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
2009 Harvard-MIT Mathematics Tournament, 1
How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?
PEN A Problems, 86
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.
1983 Austrian-Polish Competition, 2
Find all triples of positive integers $(p, q, n)$ with $p$ and $q$ prime, such that $p(p+1)+q(q+1) = n(n+1)$.
2017 F = ma, 3
A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface?
$\textbf{(A)}R\qquad
\textbf{(B)}\frac{R}{2}\qquad
\textbf{(C)}\frac{R}{4}\qquad
\textbf{(D)}\frac{3R}{8}\qquad
\textbf{(E)}\frac{R}{8}$