Found problems: 85335
2023 Thailand October Camp, 4
Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.
1995 Rioplatense Mathematical Olympiad, Level 3, 5
Consider $2n$ points in the plane. Two players $A$ and $B$ alternately choose a point on each move. After $2n$ moves, there are no points left to choose from and the game ends.
Add up all the distances between the points chosen by $A$ and add up all the distances between the points chosen by $B$. The one with the highest sum wins.
If $A$ starts the game, describe the winner's strategy.
Clarification: Consider that all the partial sums of distances between points give different numbers.
1997 Austrian-Polish Competition, 1
Let $P$ be the intersection of lines $l_1$ and $l_2$. Let $S_1$ and $S_2$ be two
circles externally tangent at $P$ and both tangent to $l_1$, and let $T_1$
and $T_2$ be two circles externally tangent at $P$ and both tangent to $l_2$.
Let $A$ be the second intersection of $S_1$ and $T_1, B$ that of $S_1$ and $T_2,
C$ that of $S_2$ and $T_1$, and $D$ that of $S_2$ and $T_2$. Show that the points $A,B,C,D$ are concyclic if and only if $l_1$ and $l_2$ are perpendicular.
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
2017 Switzerland - Final Round, 10
Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that:
$$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$
2024 Bulgaria MO Regional Round, 10.2
Given are two fixed lines that meet at a point $O$ and form an acute angle with measure $\alpha$. Let $P$ be a fixed point, internal for the angle. The points $M, N$ vary on the two lines (one point on each line) such that $\angle MPN=180^{\circ}-\alpha$ and $P$ is internal for $\triangle MON$. Show that the foot of the perpendicular from $P$ to $MN$ lies on a fixed circle.
2011 India IMO Training Camp, 3
A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be [i]balanced[/i] if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum.
$a)$ Prove that if there exist [i]balanced[/i] sets of sizes $k,j$ then also a [i]balanced[/i] set of size $k+j-1$.
$b)$ Prove that for all [i]odd[/i] $n\geq 7$ there exist a [i]balanced[/i] set of size $n$.
1998 Tournament Of Towns, 3
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.)
(G Galperin)
2011 Saudi Arabia Pre-TST, 2.3
Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.
2014 ASDAN Math Tournament, 4
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
2013 Putnam, 2
Let $S$ be the set of all positive integers that are [i]not[/i] perfect squares. For $n$ in $S,$ consider choices of integers $a_1,a_2,\dots, a_r$ such that $n<a_1<a_2<\cdots<a_r$ and $n\cdot a_1\cdot a_2\cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. For example, $2\cdot 3\cdot 6$ is a perfect square, while $2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ and $2\cdot 3\cdot 4\cdot 5$ are not, and so $f(2)=6.$ Show that the function $f$ from $S$ to the integers is one-to-one.
2013 Saint Petersburg Mathematical Olympiad, 2
if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \]
A. Khrabrov
2023 HMNT, 10
It is midnight on April $29$th, and Abigail is listening to a song by her favorite artist while staring at her clock, which has an hour, minute, and second hand. These hands move continuously. Between two consecutive midnights, compute the number of times the hour, minute, and second hands form two equal angles and no two hands overlap.
PEN P Problems, 22
Show that an integer can be expressed as the difference of two squares if and only if it is not of the form $4k+2 \; (k \in \mathbb{Z})$.
2021 Girls in Math at Yale, R2
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$. Find the maximum possible value of $A \cdot B$.
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.
2019 JBMO Shortlist, A5
Let $a, b, c, d$ be positive real numbers such that $abcd = 1$. Prove the inequality
$\frac{1}{a^3 + b + c + d} +\frac{1}{a + b^3 + c + d}+\frac{1}{a + b + c^3 + d} +\frac{1}{a + b + c + d^3} \leq \frac{a+b+c+d}{4}$
[i]Proposed by Romania[/i]
2013 Iran MO (2nd Round), 2
Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a [i]Perfect Set [/i]if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.
1991 National High School Mathematics League, 6
The figure of equation $|x-y^2|=1-|x|$ is
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi80LzQ4YjgxN2YxMjc0YTBkNzZiZjJiMTRhMjBiNDExN2I5OGZhZGY3LnBuZw==&rn=MjAwMDAwMDAwMDAwMC5wbmc=[/img]
2003 Iran MO (3rd Round), 12
There is a lamp in space.(Consider lamp a point)
Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)
2014 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle with $AB > AC$. Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$. Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$. Prove that $BQ = 2AP.$
1985 Bulgaria National Olympiad, Problem 6
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.
2022 MOAA, 4
Angeline flips three fair coins, and if there are any tails, she then flips all coins that landed tails each one more time. The probability that all coins are now heads can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2024 Korea Summer Program Practice Test, 1
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the equation
$$f(x^2+yf(x))=(1-x)f(y-x)$$
holds for all $x,y\in\mathbb{R}$.
2023 Sharygin Geometry Olympiad, 10.4
Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$.
2019 Saudi Arabia JBMO TST, 1
Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy
$11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$.