Found problems: 303
2018 USAMO, 6
Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1$.
[i]Proposed by Richard Stong[/i]
1982 USAMO, 5
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
2004 Romania Team Selection Test, 17
On a chess table $n\times m$ we call a [i]move [/i] the following succesion of operations
(i) choosing some unmarked squares, any two not lying on the same row or column;
(ii) marking them with 1;
(iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move).
We call a [i]game [/i]a succession of moves that end in the moment that we cannot make any more moves.
What is the maximum possible sum of the numbers on the table at the end of a game?
2025 USAJMO, 2
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
2013 USAMO, 2
For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1}+a_n=2^n$ for all $n\geq 4$.
1992 AIME Problems, 8
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
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.
2006 USAMO, 1
Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and
\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]
if and only if $s$ is not a divisor of $p-1$.
Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.
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.
2014 JHMMC 7 Contest, 26
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text{USAMO}$. He will need to get at least $200$ points on $\text{the AMC}$ and $\text{AIME}$ combined and get at least $21$ points on $\text{the USAMO}$ to make $\text{MOP}$. What is the minimum amount of effort he can expend to make $\text{MOP}$?
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$).
2014 Contests, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2015 USAMO, 5
Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.
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.
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.
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?