Found problems: 85335
1995 USAMO, 5
Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.
2024 CMIMC Combinatorics and Computer Science, 3
Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
[i]Proposed by Robert Trosten[/i]
2011 Saudi Arabia IMO TST, 1
Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that
$$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
1997 India Regional Mathematical Olympiad, 5
Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.
MathLinks Contest 4th, 1.1
Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that
$$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$
2010 Grand Duchy of Lithuania, 4
In the triangle $ABC$ angle $C$ is a right angle. On the side $AC$ point $D$ has been found, and on the segment $BD$ point K has been found such that $\angle ABC = \angle KAD = \angle AKD$. Prove that $BK = 2DC$.
2023 May Olympiad, 5
There are $100$ boxes that were labeled with the numbers $00$, $01$, $02$,$…$, $99$ . The numbers $000$, $001$, $002$, $…$, $999$ were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card $037$ in box $07$, but it is not allowed to place the card $156$ in box $65$.Can it happen that after placing all the cards in the boxes, there will be exactly $50$ empty boxes?
If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible
2012 May Olympiad, 1
Pablo says: “I add $2$ to my birthday and multiply the result by $2$. I add to the number obtained $4$ and multiply the result by $5$. To the new number obtained I add the number of the month of my birthday (for example, if it's June, I add $6$) and I get $342$. "
What is Pablo's birthday date? Give all the possibilities
1982 Austrian-Polish Competition, 5
Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.
1988 AMC 8, 6
$ \frac{(.2)^{3}}{(.02)^{2}}= $
$ \text{(A)}\ .2\qquad\text{(B)}\ 2\qquad\text{(C)}\ 10\qquad\text{(D)}\ 15\qquad\text{(E)}\ 20 $
2013 Princeton University Math Competition, 4
Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.
1999 Brazil Team Selection Test, Problem 3
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
2024-25 IOQM India, 17
Consider an isosceles triangle $ABC$ with sides $BC = 30, CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. The length of $PQ$ is:
2007 Junior Tuymaada Olympiad, 3
A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure:
In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.
2022 Federal Competition For Advanced Students, P2, 4
Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$.
[i](Walther Janous)[/i]
2017 Hanoi Open Mathematics Competitions, 7
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
1957 AMC 12/AHSME, 24
If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by:
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ \text{the product of the digits}\qquad
\textbf{(C)}\ \text{the sum of the digits}\qquad
\textbf{(D)}\ \text{the difference of the digits}\qquad
\textbf{(E)}\ 11$
2000 Brazil National Olympiad, 4
An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light?
2020 CCA Math Bonanza, L2.3
$3$ uncoordinated aliens launch a $3$-day attack on $4$ galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, what is $m+n$?
[i]2020 CCA Math Bonanza Lightning Round #2.3[/i]
2024 ELMO Shortlist, N4
Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$.
[i]Linus Tang[/i]
2022 Princeton University Math Competition, B2
A pair $(f,g)$ of degree $2$ real polynomials is called [i]foolish[/i] if $f(g(x)) = f(x) \cdot g(x)$ for all real $x.$ How many positive integers less than $2023$ can be a root of $g(x)$ for some foolish pair $(f,g)$?
MOAA Accuracy Rounds, 2021.6
Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by William Yue[/i]
2009 Austria Beginners' Competition, 1
A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is $40$. Which values are possible for the sum of the red numbers?
(G. Kirchner, University of Innsbruck)
2020 Harvard-MIT Mathematics Tournament, 4
Given an $8\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?
[i]Proposed by James Lin.[/i]