Found problems: 85335
2015 APMO, 2
Let $S = \{2, 3, 4, \ldots\}$ denote the set of integers that are greater than or equal to $2$. Does there exist a function $f : S \to S$ such that \[f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2008 Dutch IMO TST, 2
Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game.
Prove that Johan can always get a score that is at least as high as Julian’s.
2009 Princeton University Math Competition, 1
You have an unlimited supply of monominos, dominos, and L-trominos. How many ways, in terms of $n$, can you cover a $2 \times n$ grid with these shapes? Please note that you do [i]NOT[/i] have to use all the shapes. Also, you are allowed to [i]rotate[/i] any of the pieces, so they do not have to be aligned exactly as they are in the diagram below.
[asy]
pen db = rgb(0,0,0.5); real r = 0.08; pair s1 = (3,0), s2 = 2*s1;
fill(unitsquare, db); fill(shift(s1)*unitsquare, db); fill(shift(s1-(0,1+r))*unitsquare, db); fill(shift(s2)*unitsquare, db); fill(shift(s2-(0,1+r))*unitsquare, db); fill(shift(s2+(1+r,-1-r))*unitsquare, db);
[/asy]
2020 AMC 12/AHSME, 14
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$
$\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$
2014 Taiwan TST Round 3, 1
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]
2022 Princeton University Math Competition, A4 / B6
Let $C_n$ denote the $n$-dimensional unit cube, consisting of the $2^n$ points $$\{(x_1, x_2, \ldots, x_n) \mid x_i \in \{0, 1\} \text{ for all } 1 \le i \le n\}.$$ A tetrahedron is [i]equilateral[/i] if all six side lengths are equal. Find the smallest positive integer $n$ for which there are four distinct points in $C_n$ that form a non-equilateral tetrahedron with integer side lengths.
2010 Math Prize for Girls Olympiad, 2
Prove that for every positive integer $n$, there exist integers $a$ and $b$ such that $4a^2 + 9b^2 - 1$ is divisible by $n$.
2014 Czech-Polish-Slovak Junior Match, 3
Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.
2013 F = Ma, 15
A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water?
$\textbf{(A) } 0.25\\
\textbf{(B) } 0.33\\
\textbf{(C) } 0.5\\
\textbf{(D) } 0.67\\
\textbf{(E) } 0.75$
2024 UMD Math Competition Part I, #24
Let $n\ge3$ be an integer. A regular $n$-gon $P$ is given. We randomly select three distinct vertices of $P$. The probability that these three vertices form an isosceles triangle is $1/m$, where $m$ is an integer. How many such integers $n\le 2024$ are there?
\[\rm a. ~674\qquad \mathrm b. ~675\qquad \mathrm c. ~682 \qquad\mathrm d. ~684\qquad\mathrm e. ~685\]
2003 USAMO, 2
A convex polygon $\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
2009 Brazil Team Selection Test, 2
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]
2016 Mathematical Talent Reward Programme, SAQ: P 3
Prove that for any positive integer $n$ there are $n$ consecutive composite numbers all less than $4^{n+2}$.
2004 Romania National Olympiad, 1
Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$.
Prove that \[ FB_1 + FB_2 + \ldots + FB_n > np . \]
[i]Calin Popescu[/i]
2024 Thailand October Camp, 3
Let triangle \( ABC \) be an acute-angled triangle. Square \( AEFB \) and \( ADGC \) lie outside triangle \( ABC \). \( BD \) intersects \( CE \) at point \( H \), and \( BG \) intersects \( CF \) at point \( I \). The circumcircle of triangle \( BFI \) intersects the circumcircle of triangle \( CGI \) again at point \( K \). Prove that line segment \( HK \) bisects \( BC \).
2023 ISL, N5
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2005 All-Russian Olympiad, 1
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.
2016 Harvard-MIT Mathematics Tournament, 4
Let $n > 1$ be an odd integer. On an $n \times n$ chessboard the center square and four corners are deleted.
We wish to group the remaining $n^2-5$ squares into $\frac12(n^2-5)$ pairs, such that the two squares in each pair intersect at exactly one point
(i.e.\ they are diagonally adjacent, sharing a single corner).
For which odd integers $n > 1$ is this possible?
2024 Ukraine National Mathematical Olympiad, Problem 2
For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes.
[i]Proposed by Anton Trygub[/i]
2016 Purple Comet Problems, 27
A container the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the container. Find the volume of the liquid in the container.
[center][img]https://snag.gy/CdvpUq.jpg[/img][/center]
2013 Estonia Team Selection Test, 6
A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.
2012 Purple Comet Problems, 29
Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.
2005 Tournament of Towns, 3
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)
[i](5 points)[/i]
1957 AMC 12/AHSME, 23
The graph of $ x^2 \plus{} y \equal{} 10$ and the graph of $ x \plus{} y \equal{} 10$ meet in two points. The distance between these two points is:
$ \textbf{(A)}\ \text{less than 1} \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ \sqrt{2}\qquad
\textbf{(D)}\ 2\qquad
\textbf{(E)}\ \text{more than 2}$
2015 Purple Comet Problems, 16
\[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$.