Found problems: 85335
2022-23 IOQM India, 3
In a trapezoid $ABCD$, the internal bisector of angle $A$ intersects the base $BC$(or its extension) at the point $E$. Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$. Find the angle $DAE$ in degrees, if $AB:MP=2$.
2011 IFYM, Sozopol, 1
Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.
2011 QEDMO 10th, 9
Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?
2001 Pan African, 3
Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$. A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$, which is the symmetric point of $P_0$ with respect to $A$. From $P_1$, the grasshopper jumps to $P_2$, which is the symmetric point of $P_1$ with respect to $B$. Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$, and so on. Compare the distance $P_0P_1$ and $P_0P_n$. $n \in N$.
2021 Israel TST, 4
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
2014 Belarus Team Selection Test, 1
Let $I$ be the incenter of a triangle $ABC$. The circle passing through $I$ and centered at $A$ meets the circumference of the triangle $ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the incircle of the triangle $ABC$.
(I. Kachan)
2008 Grigore Moisil Intercounty, 2
Determine the natural numbers a, b, c s.t. :
$ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $
The challenge here is to come up with as basic solution as possible.
PEN Q Problems, 6
Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.
2023 Pan-American Girls’ Mathematical Olympiad, 6
Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.
2017 SDMO (High School), 5
There are $n$ dots on the plane such that no three dots are collinear. Each dot is assigned a $0$ or a $1$. Each pair of dots is connected by a line segment. If the endpoints of a line segment are two dots with the same number, then the segment is assigned a $0$. Otherwise, the segment is assigned a $1$. Find all $n$ such that it is possible to assign $0$'s and $1$'s to the $n$ dots in a way that the corresponding line segments are assigned equally many $0$'s as $1$'s.
2024 CMIMC Integration Bee, 3
\[\int_0^1 \frac{\log(x)}{\sqrt x}\mathrm dx\]
[i]Proposed by Robert Trosten[/i]
1998 Poland - Second Round, 3
If $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ are nonnegative real numbers satisfying $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f \equal{} 1$ and $ ace \plus{} bdf \geq \frac {1}{108}$, then prove that
\[ abc \plus{} bcd \plus{} cde \plus{} de f \plus{} efa \plus{} fab \leq \frac {1}{36}
\]
2011 AMC 12/AHSME, 22
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320 \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$
Kyiv City MO 1984-93 - geometry, 1990.11.1
Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.
2008 Puerto Rico Team Selection Test, 1
Given a $ 1 \times 25$ rectangle divided into $ 25$ "boxes" ($ 1 \times 1$), is it possible to write integers $ 1$ to $ 25$ so that the sum of any two adjacent "boxes" is a perfect square?
2022 Yasinsky Geometry Olympiad, 2
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi)
1951 Polish MO Finals, 2
What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?
1967 All Soviet Union Mathematical Olympiad, 090
In the sequence of the natural (i.e. positive integers) numbers every member from the third equals the absolute value of the difference of the two previous. What is the maximal possible length of such a sequence, if every member is less or equal to $1967$?
2024 Alborz Mathematical Olympiad, P2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$
Proposed by Heidar Shushtari
2019-2020 Fall SDPC, 2
Consider a function $f: \mathbb{Z} \rightarrow \mathbb{Z}$. We call an integer $a$ [i]spanning[/i] if for all integers $b \neq a$, there exists a positive integer $k$ with $f^k(a)=b$. Find, with proof, the maximum possible number of [i]spanning[/i] numbers of $f$.
Note: $\mathbb{Z}$ represents the set of all integers, so $f$ is a function from the set of integers to itself. $f^k(a)$ is defined as $f$ applied $k$ times to $a$.
2017 IMO Shortlist, A5
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
2024 Oral Moscow Geometry Olympiad, 2
The bisector $BL$ was drawn in the triangle $ABC$. Let the points $I_1$ and $I_2$ be centers of the circles inscribed in the triangles $ABL$ and $CBL$, and the points $J_1$ and $J_2$ be centers of the excircles of these triangles touching the side $BL$. Prove that the points $I_1$, $I_2$, $J_1$ and $J_2$ lie on the same circle.
2023 Malaysian IMO Team Selection Test, 3
Let $ABC$ be an acute triangle with $AB\neq AC$. Let $D, E, F$ be the midpoints of the sides $BC$, $CA$, and $AB$ respectively, and $M, N$ be the midpoints of minor arc $BC$ not containing $A$ and major arc $BAC$ respectively. Suppose $W, X, Y, Z$ are the incenter, $D$-excenter, $E$-excenter, and $F$-excenter of triangle $DEF$ respectively.
Prove that the circumcircles of the triangles $ABC$, $WNX$, $YMZ$ meet at a common point.
[i]Proposed by Ivan Chan Kai Chin[/i]
2017 Sharygin Geometry Olympiad, P4
A triangle $ABC$ is given. Let $C\ensuremath{'}$ be the vertex of an isosceles triangle $ABC\ensuremath{'}$ with $\angle C\ensuremath{'} = 120^{\circ}$ constructed on the other side of $AB$ than $C$, and $B\ensuremath{'}$ be the vertex of an equilateral triangle $ACB\ensuremath{'}$ constructed on the same side of $AC$ as $ABC$. Let $K$ be the midpoint of $BB\ensuremath{'}$
Find the angles of triangle $KCC\ensuremath{'}$.
[i]Proposed by A.Zaslavsky[/i]
2001 Chile National Olympiad, 1
$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$?
$\bullet$ In how many of these triangles is the perimeter divisible by $3$?