Found problems: 85335
2007 Indonesia MO, 8
Let $ m$ and $ n$ be two positive integers. If there are infinitely many integers $ k$ such that $ k^2\plus{}2kn\plus{}m^2$ is a perfect square, prove that $ m\equal{}n$.
2017 AIME Problems, 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, as shown, is $\tfrac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
[asy]
size(5cm);
pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);
real t = .385, s = 3.5*t-1;
pair R = A*t+B*(1-t), P=B*s;
pair Q = dir(-60) * (R-P) + P;
fill(P--Q--R--cycle,gray);
draw(A--B--C--A^^P--Q--R--P);
dot(A--B--C--P--Q--R);
[/asy]
2021 Purple Comet Problems, 17
For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2013 Junior Balkan Team Selection Tests - Romania, 4
Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that lines $AD, EF$ and $H_1H_2$ are concurrent.
2011 Tournament of Towns, 5
In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
2024 Putnam, A2
For which real polynomials $p$ is there a real polynomial $q$ such that
\[
p(p(x))-x=(p(x)-x)^2q(x)
\]
for all real $x$?
1999 AIME Problems, 12
The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.
Novosibirsk Oral Geo Oly VIII, 2019.7
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
2020 Iran MO (3rd Round), 1
find all functions from the reals to themselves. such that for every real $x,y$.
$$f(y-f(x))=f(x)-2x+f(f(y))$$
2020-21 KVS IOQM India, 24
Two circles $S_1$ and $S_2$, of radii $6$ units and $3$ units respectively, are tangent to each other, externally. Let $AC$ and $BD$ be their direct common tangents with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest Integer.
2020 LMT Spring, 8
Let $a,b$ be real numbers satisfying $a^{2} + b^{2} = 3ab = 75$ and $a>b$. Compute $a^{3}-b^{3}$.
2023 pOMA, 5
Let $n\ge 2$ be a positive integer, and let $P_1P_2\dots P_{2n}$ be a polygon with $2n$ sides such that no two sides are parallel. Denote $P_{2n+1}=P_1$. For some point $P$ and integer $i\in\{1,2,\ldots,2n\}$, we say that $i$ is a $P$-good index if $PP_{i}>PP_{i+1}$, and that $i$ is a $P$-bad index if $PP_{i}<PP_{i+1}$.
Prove that there's a point $P$ such that the number of $P$-good indices is the same as the number of $P$-bad indices.
2001 Federal Competition For Advanced Students, Part 2, 3
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.
2012 Iran Team Selection Test, 1
Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity.
[i]Proposed by Mr.Etesami[/i]
2021 AIME Problems, 3
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products
$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$$
is divisible by $3$.
1962 All Russian Mathematical Olympiad, 016
Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.
2004 Cuba MO, 8
Determine all functions $f : R_+ \to R_+$ such that:
a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$
b) $f(2) = 0$
c) $f(x) \ne 0$ for $0 \le x < 2$.
2012 Federal Competition For Advanced Students, Part 2, 2
Solve over $\mathbb{Z}$:
\[ x^4y^3(y-x)=x^3y^4-216 \]
1987 Austrian-Polish Competition, 5
The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.
1961 Polish MO Finals, 5
Four lines intersecting at six points form four triangles. Prove that the circles circumscribed around out these triangles have a common point.
2023 Germany Team Selection Test, 2
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
2023 USEMO, 6
Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$) such that there exist complex numbers $r_1$, $\dots$, $r_n$, not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $m$, find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list]
[i]Kaixin Wang[/i]
1969 IMO, 4
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
1992 IMO Longlists, 8
Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation
\[f(f(x)) + af(x) = b(a + b)x.\]
Prove that there exists a unique solution of this equation.
2020 Iran Team Selection Test, 1
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.
[i]Proposed by Masud Shafaie[/i]