Found problems: 85335
2021 Junior Balkan Team Selection Tests - Moldova, 5
Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.
2005 CHKMO, 4
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
2003 Junior Macedonian Mathematical Olympiad, Problem 1
Show that for every positive integer $n$ the number $7^n-1$ is not divisible by $6^n-1$.
2009 Germany Team Selection Test, 2
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
2020 SMO, 3
Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and incircle $\omega$. Two points $X$ and $Y$ are chosen on minor arcs $AB$ and $AC$, respectively, of the circumcircle of triangle $\triangle ABC$ such that $XY$ is tangent to $\omega$ at $P$ and $\overline{XY}\perp \overline{AI}$. Let $\omega$ be tangent to sides $AC$ and $AB$ at $E$ and $F$, respectively. Denote the intersection of lines $XF$ and $YE$ as $T$.
Prove that if the circumcircles of triangles $\triangle TEF$ and $\triangle ABC$ are tangent at some point $Q$, then lines $PQ$, $XE$, and $YF$ are concurrent.
[i]Proposed by Andrew Wen[/i]
2016 ASDAN Math Tournament, 6
Suppose we have $3$ baskets and $4$ distinguishable balls. Each ball is placed into a randomly selected basket. Compute the probability that the basket with the most balls has at least $3$ balls.
2022 Assara - South Russian Girl's MO, 2
Numbers $1, 2, 3, . . . , 100$ are arranged in a circle in some order. A [i]good pair[/i] is a pair of numbers of the same parity, between which there are exactly $3$ numbers. What is the smallest possible number good pairs?
2020 Simon Marais Mathematics Competition, A1
There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments.
Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point.
[i]An interior point of a line segment is a point of the line segment that is not one of the two endpoints.[/i]
2004 Bulgaria Team Selection Test, 1
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.
2011 Bogdan Stan, 4
Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct,
[b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal.
[b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational.
[i]Marin Tolosi[/i]
2008 Gheorghe Vranceanu, 1
Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $
2019 USMCA, 13
The infinite sequence $a_0,a_1,\ldots$ is given by $a_1=\frac12$, $a_{n+1} = \sqrt{\frac{1+a_n}{2}}$. Determine the infinite product $a_1a_2a_3\cdots$.
2016 AMC 10, 7
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$
2023 Rioplatense Mathematical Olympiad, 1
An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice:
$$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$
Determine all poli-pythagorean integers.
2021 Turkey MO (2nd round), 5
There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.
2009 Vietnam Team Selection Test, 1
Let $ a,b,c$ be positive numbers.Find $ k$ such that:
$ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$
1951 AMC 12/AHSME, 46
$ AB$ is a fixed diameter of a circle whose center is $ O$. From $ C$, any point on the circle, a chord $ CD$ is drawn perpendicular to $ AB$. Then, as $ C$ moves over a semicircle, the bisector of angle $ OCD$ cuts the circle in a point that always:
$ \textbf{(A)}\ \text{bisects the arc } AB \qquad\textbf{(B)}\ \text{trisects the arc } AB \qquad\textbf{(C)}\ \text{varies}$
$ \textbf{(D)}\ \text{is as far from }AB \text{ as from } D \qquad\textbf{(E)}\ \text{is equidistant from }B \text{ and } C$
2007 Romania National Olympiad, 2
Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]
1951 Poland - Second Round, 1
In a right triangle $ ABC $, the altitude $ CD $ is drawn from the vertex of the right angle $ C $ and a circle is inscribed in each of the triangles $ ABC $, $ ACD $ and $ BCD $. Prove that the sum of the radii of these circles equals the height $ CD $.
2016 Saudi Arabia IMO TST, 2
Find all pairs of polynomials $P(x),Q(x)$ with integer coefficients such that $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$ for all real numbers $x$
2000 All-Russian Olympiad Regional Round, 11.2
The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?
2024 OMpD, 3
For each positive integer \( n \), let \( f(n) \) be the number of ordered triples \( (a, b, c) \) such that \( a, b, c \in \{1, 2, \ldots, n\} \) and that the two roots (possibly equal) of the quadratic equation \( ax^2 + bx + c = 0 \) are both integers.
(a) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) > C \cdot n \).
(b) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) < C \cdot n^{\frac{2025}{2024}} \).
1989 IMO, 2
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
2015 BMT Spring, Tie 3
The permutohedron of order $3$ is the hexagon determined by points $(1, 2, 3)$, $(1, 3, 2)$, $(2, 1, 3)$, $(2, 3, 1)$, $(3, 1, 2)$, and $(3, 2, 1)$. The pyramid determined by these six points and the origin has a unique inscribed sphere of maximal volume. Determine its radius.
2013 Indonesia MO, 2
Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.