Found problems: 85335
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
2024 Harvard-MIT Mathematics Tournament, 31
Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated.
Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$
2009 Mexico National Olympiad, 2
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:
$\bullet$ If $p$ is a prime number, we place it in box $1$.
$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.
Find all positive integers $n$ that are placed in the box labeled $n$.
2015 South Africa National Olympiad, 3
We call a divisor $d$ of a positive integer $n$ [i]special[/i] if $d + 1$ is also a divisor of $n$. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.
2023 LMT Fall, 13
Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown.
[img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img]
On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?
2007 IMO Shortlist, 1
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
2002 All-Russian Olympiad Regional Round, 8.4
Given a triangle $ABC$ with pairwise distinct sides. on his on the sides, regular triangles $ABC_1$, $BCA_1$, $CAB_1$. are constructed externally. Prove that triangle $A_1B_1C_1$ cannot be regular.
2020 Online Math Open Problems, 4
Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$.
[i]Proposed by Brandon Wang[/i]
2012 Dutch IMO TST, 4
Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.
2011 All-Russian Olympiad, 3
Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.
[i]A. Golovanov[/i]
1999 May Olympiad, 2
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.
1970 IMO Longlists, 55
A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
2013 NIMO Problems, 8
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan Chen[/i]
2000 All-Russian Olympiad Regional Round, 10.7
In a convex quadrilateral $ABCD$ we draw the bisectors $\ell_a$, $\ell_b$, $\ell_c$, $\ell_d$ of external angles $A$, $B$, $C$, $D$ respectively. The intersection points of the lines $\ell_a$ and $\ell_b$, $\ell_b$ and $\ell_c$, $\ell_c$ and $\ell_d$, $\ell_d$ and $\ell_a$ are designated by $K$, $L$, $M$, $N$. It is known that $3$ perpendiculars drawn from $K$ on $AB$, from $L$ om $BC$, from $M$ on $CD$ intersect at one point. Prove that the quadrilateral $ABCD$ is cyclic.
2003 Greece JBMO TST, 2
Calculate if $n\in N$ with $n>2$ the value of
$$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$
2022 JHMT HS, 5
Consider an array of white unit squares arranged in a rectangular grid with $59$ rows of unit squares and $c$ columns of unit squares, for some positive integer $c$. What is the smallest possible value of $c$ such that, if we shade exactly $25$ unit squares in each column black, then there must necessarily be some row with at least $18$ black unit squares?
1949-56 Chisinau City MO, 44
Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.
1991 National High School Mathematics League, 1
Set $S=\{1,2,\cdots,n\}$. $A$ is an increasing arithmetic sequence (at least two numbers), and all numbers are in $S$. Also, we can't add any number in $S$ to $A$ without changing its tolerance. Find the number of such sequence $A$.
2017 Peru IMO TST, 4
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
2022 Malaysia IMONST 2, 2
It is known that there are $n$ integers $a_1, a_2, \cdots, a_n$ such that
$$a_1 + a_2 + \cdots + a_n = 0 \qquad \text{and} \qquad a_1 \times a_2 \times \cdots \times a_n = n.$$
Determine all possible values of $n$.
2014 Poland - Second Round, 2.
Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.
1986 Putnam, A4
A [i]transversal[/i] of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
(a) Each entry $\alpha_{i,j}$ of $A$ is in the set $\{-1,0,1\}$.
(b) The sum of the $n$ entries of a transversal is the same for all transversals of $A$.
An example of such a matrix $A$ is
\[
A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0
\end{array}
\right).
\]
Determine with proof a formula for $f(n)$ of the form
\[
f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,
\]
where the $a_i$'s and $b_i$'s are rational numbers.
2002 India IMO Training Camp, 3
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?
2018 Thailand TST, 2
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?
2000 China Team Selection Test, 3
For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that:
a.) $N_a$ is odd;
b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.