Found problems: 275
2010 Contests, 1
Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
2004 AMC 10, 1
You and five friends need to raise $ \$1500$ in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
$ \textbf{(A)}\ 250\qquad
\textbf{(B)}\ 300\qquad
\textbf{(C)}\ 1500\qquad
\textbf{(D)}\ 7500\qquad
\textbf{(E)}\ 9000$
2002 USAMO, 2
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
2012 USAJMO, 2
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
2008 USAMO, 6
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
2021 USAJMO, 4
Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a [i]move[/i], Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?
(A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.)
2013 USAMO, 5
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
2015 USAJMO, 2
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
2014 NIMO Problems, 8
Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States).
What is $a^2+b^2+c^2$?
041. The Gentleman's Alliance Cross
042. Glutamine (an amino acid)
051. Grant Nelson and Norris Windross
052. A compact region at the center of a galaxy
061. The value of \verb+'wat'-1+. (See \url{https://www.destroyallsoftware.com/talks/wat}.)
062. Threonine (an amino acid)
071. Nintendo Gamecube
072. Methane and other gases are compressed
081. A prank or trick
082. Three carbons
091. Australia's second largest local government area
092. Angoon Seaplane Base
101. A compressed archive file format
102. Momordica cochinchinensis
111. Gentaro Takahashi
112. Nat Geo
121. Ante Christum Natum
122. The supreme Siberian god of death
131. Gnu C Compiler
132. My TeX Shortcut for $\angle$.
1970 AMC 12/AHSME, 33
Find the sum of the digits of all numerals in the sequence $1,2,3,4,\cdots ,10000$.
$\textbf{(A) }180,001\qquad\textbf{(B) }154,756\qquad\textbf{(C) }45,001\qquad\textbf{(D) }154,755\qquad \textbf{(E) }270,001$
2014 South East Mathematical Olympiad, 3
In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$
PEN M Problems, 32
In an increasing infinite sequence of positive integers, every term starting from the $2002$-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
2022 USAJMO, 6
Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n \\ b_{n+1} &= b_n^2 + 2c_na_n \\ c_{n+1} &= c_n^2 + 2a_nb_n\end{align*}for all nonnegative integers $n.$
Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$
2000 Korea - Final Round, 2
Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$,
\[f(x^2-y^2)=(x-y)(f(x)+f(y))\]
2005 District Olympiad, 1
Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10:
\[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]
2016 Switzerland Team Selection Test, Problem 6
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
2016 USAJMO, 2
Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.
[i]Proposed by Evan Chen[/i]
2015 USAJMO, 6
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?
2019 USAMO, 5
Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps.
[i]Proposed by Yannick Yao[/i]
1976 USAMO, 1
(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with [i]any[/i] such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color.
(b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.
2011 USAJMO, 5
Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$. Prove that $BE$ bisects $AC$.
1985 USAMO, 1
Determine whether or not there are any positive integral solutions of the simultaneous equations
\begin{align*}x_1^2+x_2^2+\cdots+x_{1985}^2&=y^3,\\
x_1^3+x_2^3+\cdots+x_{1985}^3&=z^2\end{align*}
with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.
2015 USAJMO, 5
Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\overline{BD}$ such that $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there exists a point $Y$ on segment $\overline{AC}$ such that $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$.
2019 USAJMO, 5
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
[i]Proposed by Ricky Liu[/i]