Found problems: 85335
2010 Silk Road, 3
For positive real numbers $a, b, c, d,$ satisfying the following conditions:
$a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$
1984 Austrian-Polish Competition, 7
A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.
1986 IMO Longlists, 43
Three persons $A,B,C$, are playing the following game:
A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$.
For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)
2016 ASDAN Math Tournament, 5
Find
$$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$
2011 Croatia Team Selection Test, 2
There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$
2018 India PRMO, 17
Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D, AB = DE = 17, BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?
2024 ELMO Shortlist, A2
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
2010 Junior Balkan Team Selection Tests - Romania, 4
The plan considers $51$ points of integer coordinates, so that the distances between any two points are natural numbers. Show that at least $49\%$ of the distances are even.
2021 239 Open Mathematical Olympiad, 1
You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?
2020 AMC 10, 25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$
1992 Mexico National Olympiad, 2
Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?
2022 Sharygin Geometry Olympiad, 13
Eight points in a general position are given in the plane. The areas of all $56$ triangles with vertices at these points are written in a row. Prove that it is possible to insert the symbols "$+$" and "$-$" between them in such a way
that the obtained sum is equal to zero.
2003 Romania Team Selection Test, 5
Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials.
[i]Mihai Piticari[/i]
2018 NZMOC Camp Selection Problems, 2
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$
2016 Purple Comet Problems, 13
In $\triangle$ABC shown below, AB = AC, AF = EF, and EH = CH = DH = GH = DG = BG. Also,
∠CHE = ∠F GH. Find the degree measure of ∠BAC.
[center][img]https://i.snag.gy/ZyxQVX.jpg[/img][/center]
2018 Turkey MO (2nd Round), 5
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which
$$
({\rm gcd}(a_i,a_j))^2|a_k.
$$
2003 Baltic Way, 1
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil
\[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
2023 Macedonian Team Selection Test, Problem 2
Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that
$$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$
[i]Authored by Nikola Velov[/i]
1976 Miklós Schweitzer, 1
Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive.
[i]L. Posa[/i]
2013 NIMO Problems, 1
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
[i]Proposed by Evan Chen[/i]
2014 PUMaC Geometry A, 6
$\triangle ABC$ has side lengths $AB=15$, $BC=34$, and $CA=35$. Let the circumcenter of $ABC$ be $O$. Let $D$ be the foot of the perpendicular from $C$ to $AB$. Let $R$ be the foot of the perpendicular from $D$ to $AC$, and let $W$ be the perpendicular foot from $D$ to $BC$. Find the area of quadrilateral $CROW$.
2005 iTest, 9
Find the area of the triangle with vertices of $(1,2)$, $(1,10)$, and $(5, 5)$.
2021 Science ON all problems, 1
Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$.
$\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$.
$\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that
$$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$
Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$.
[i] (Stefan Obadă)[/i]
2011 Tournament of Towns, 2
Several guests at a round table are eating from a basket containing $2011$ berries. Going in clockwise direction, each guest has eaten either twice as many berries as or six fewer berries than the next guest. Prove that not all the berries have been eaten.
1996 May Olympiad, 2
Joining $15^3 = 3375$ cubes of $1$ cm$^3$, bodies with a volume of $3375$ cm$^3$ can be built. Indicate how two bodies $A$ and $B$ are constructed with $3375$ cubes each and such that the lateral surface of $B$ is $10$ times the lateral surface of $A$.