Found problems: 85335
2002 Junior Balkan Team Selection Tests - Romania, 3
A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.
2025 Israel TST, P1
Let \(\mathcal{F}\) be a family of functions from \(\mathbb{R}^+ \to \mathbb{R}^+\). It is known that for all \( f, g \in \mathcal{F} \), there exists \( h \in \mathcal{F} \) such that for all \( x, y \in \mathbb{R}^+ \), the following equation holds:
\[
y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy)
\]
Prove that for all \( f \in \mathcal{F} \) and all \( x \in \mathbb{R}^+ \), the following identity is satisfied:
\[
f\left(\frac{x}{f(x)}\right) = 1.
\]
2016 HMNT, 4
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
2009 Indonesia TST, 3
Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that
\[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y)
\]
for all $ x,y \in \mathbb{R}$.
2010 Contests, 1
A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$.
[img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]
2025 AIME, 4
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.
2021 Princeton University Math Competition, A7
Let $ABC$ be a triangle with side lengths $AB = 13$, $AC = 17$, and $BC = 20$. Let $E, F$ be the feet of the altitudes from $B$ onto $AC$ and $C$ onto $AB$, respectively. Let $P$ be the second intersection of the circumcircles of $ABC$ and $AEF$. Suppose that $AP$ can be written as $\frac{a \sqrt{b}}{c}$ where $a, c$ are relatively prime and $b$ is square-free. Compute $a$.
1999 Romania Team Selection Test, 2
Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.
2011 Tournament of Towns, 1
An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?
2014 ELMO Shortlist, 2
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]
2021 Regional Olympiad of Mexico West, 3
The sequence of real numbers $a_1, a_2, a_3, ...$ is defined as follows: $a_1 = 2019$, $a_2 = 2020$, $a_3 = 2021$ and for all $n \ge 1$
$$a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.$$
Show that this sequence does not contain numbers of the form $m^6$ where $m$ is a positive integer.
2019 Math Prize for Girls Problems, 13
Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.
2003 Chile National Olympiad, 2
Find all primes $p, q$ such that $p + q = (p-q)^3$.
2012 Iran MO (3rd Round), 2
Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition:
\[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\]
[i]Proposed by Mohammad Gharakhani[/i]
2012 National Olympiad First Round, 2
Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers.
$ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$
Kvant 2020, M2631
There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that:
[list=a]
[*]one of its angles is not greater than $60^\circ{}$;
[*]one of its angles is at least $120^\circ$.
[/list]
[i]Maxim Didin[/i]
1996 Korea National Olympiad, 7
Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$
2021 Final Mathematical Cup, 2
The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .
2023 SG Originals, Q5
Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^{o}$.
2010 Sharygin Geometry Olympiad, 4
In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.
2013-2014 SDML (High School), 8
Twenty-four congruent squares are arranged as shown in the figure. In how many ways can we select $12$ of the squares so that no two are diagonally adjacent? Directly adjacent spaces are acceptable.
2004 AMC 10, 8
Minneapolis-St. Paul International Airport is $ 8$ miles southwest of downtown St. Paul and $ 10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$ \textbf{(A)}\ 13\qquad
\textbf{(B)}\ 14\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 16\qquad
\textbf{(E)}\ 17$
1988 Romania Team Selection Test, 12
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.
[i]Laurentiu Panaitopol[/i]
2002 Korea - Final Round, 1
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
2022 Saint Petersburg Mathematical Olympiad, 1
The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.