Found problems: 85335
2011 Princeton University Math Competition, A1
Find, with proof, all triples of positive integers $(x,y,z)$ satisfying the equation $3^x - 5^y = 4z^2$.
2023 Korea - Final Round, 5
Given a positive integer $n$, there are $n$ boxes $B_1,...,B_n$. The following procedure can be used to add balls.
$$\text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j.$$
For positive integers $x_1,...,x_n$ let $f(x_1,...,x_n)$ be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with $x_i$ balls for $B_i(i=1,...,n)$. Find the largest possible value of $f(x_1,...,x_n)$. (If $x_1,...,x_n$ are all multiples of 3, $f(x_1,...,x_n)=0$.)
2017 Harvard-MIT Mathematics Tournament, 18
Let $ABCD$ be a quadrilateral with side lengths $AB = 2$, $BC = 3$, $CD = 5$, and $DA = 4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$?
2015 Online Math Open Problems, 7
Define sequence $\{a_n\}$ as following: $a_0=0$, $a_1=1$, and $a_{i}=2a_{i-1}-a_{i-2}+2$ for all $i\geq 2$. Determine the value of $a_{1000}.$
[i] Proposed by Yannick Yao [/i]
2019 BMT Spring, 1
How many integers $ x $ satisfy $ x^2 - 9x + 18 < 0 $?
2016 HMNT, 2
Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?
2009 Harvard-MIT Mathematics Tournament, 4
How many functions $f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}$ satisfy $f(f(x)) = f(x)$ for all $x\in\{ 1,2, 3, 4, 5\}$?
1990 India Regional Mathematical Olympiad, 4
Find the remainder when $2^{1990}$ is divided by $1990.$
2019 China Team Selection Test, 5
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
2007 Germany Team Selection Test, 3
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
2019 Macedonia Junior BMO TST, 2
Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.
2022 Iran Team Selection Test, 12
suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that
$\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$
$\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$
$\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$
Proposed by Matin Yousefi
1975 USAMO, 2
Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]
1991 Romania Team Selection Test, 2
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
2021 Polish MO Finals, 3
Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be any point on $\omega$ different than the verticies of the triangle.
Line $AP$ intersects $BC$ at $D$, $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$. Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$, $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$. Let $Q$ be the circumcenter of triangle $XYZ$. Prove that all possible points $Q$, corresponding to different positions of $P$ lie on one circle.
1986 Polish MO Finals, 5
There is a chess tournament with $2n$ players ($n > 1$). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most $n^2$ matches. Conversely, show that if there are at most $n^2$ matches, then it is possible to arrange them so that we cannot find three players who all play each other.
2014 Contests, 3
Is there a convex pentagon in which each diagonal is equal to a side?
1985 Greece National Olympiad, 2
Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.
2021 AMC 10 Fall, 8
A two-digit positive integer is said to be [i]cuddly[/i] if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2005 USAMTS Problems, 5
Given acute triangle $\triangle ABC$ in plane $P$, a point $Q$ in space is defined such that $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$ Point $X$ is the point in plane $P$ such that $QX$ is perpendicular to plane $P$. Given $\angle ABC = 40^\circ$ and $\angle ACB = 75^\circ,$ find $\angle AXC.$
2010 HMNT, 7
$ABC$ is a right triangle with $\angle A = 30^o$ and circumcircle $O$. Circles $\omega_1$, $\omega_2$, and $\omega_3$ lie outside $ABC$ and are tangent to $O$ at $T_1$, $T_2$, and $T_3$ respectively and to $AB$, $BC$, and $CA$ at $S_1$, $S_2$, and $S_3$, respectively. Lines $T_1S_1$, $T_2S_2$, and $T_3S_3$ intersect $O$ again at $A'$, $B'$, and $C'$, respectively. What is the ratio of the area of $A'B'C'$ to the area of $ABC$?
2015 Putnam, A2
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
2018 Serbia JBMO TST, 4
Two players are playing the following game.
They are alternatively putting blue and red coins on the board $2018$ by $2018$. If first player creates $n$ blue coins in a row or column, he wins. Second player wins if he can prevent it. Who will win if:
$a)n=4$;
$b)n=5$?
Note: first player puts only blue coins, and second only red.
1969 AMC 12/AHSME, 16
When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:
$\textbf{(A) }\tfrac12k(k-1)\qquad
\textbf{(B) }\tfrac12k(k+1)\qquad
\textbf{(C) }2k-1\qquad
\textbf{(D) }2k\qquad
\textbf{(E) }2k+1$
2020 Bundeswettbewerb Mathematik, 2
Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves.
Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves.
For which values of $n$ do they need the same number of moves?