Found problems: 85335
1991 National High School Mathematics League, 1
The number of regular triangles that three apexes are among eight vertex of a cube is
$\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$
2021 Canadian Junior Mathematical Olympiad, 3
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$, $|AB|>|CD|$, and equal edges $|AD|=|BC|$. Let $I$ be the center of the circle tangent to lines $AB$, $AC$ and $BD$, where $A$ and $I$ are on opposite sides of $BD$. Let $J$ be the center of the circle tangent to lines $CD$, $AC$ and $BD$, where $D$ and $J$ are on opposite sides of $AC$. Prove that $|IC|=|JB|$.
2012 Abels Math Contest (Norwegian MO) Final, 3b
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
2019 Bulgaria EGMO TST, 3
In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded)
\[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \]
as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.
2006 Oral Moscow Geometry Olympiad, 1
The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$.
(A Zaslavsky)
2016 China Team Selection Test, 5
Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other.
Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.
2025 Bangladesh Mathematical Olympiad, P6
Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively and let $I$ be the center of that circle. Furthermore, let $P$ be the foot of the perpendicular from point $I$ to line $AD$ and let $M$ be the midpoint of $DE$. If $N$ is the intersection point of $PM$ and $AC$, prove that $DN \parallel EF$.
2010 Benelux, 2
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
1962 AMC 12/AHSME, 37
$ ABCD$ is a square with side of unit length. Points $ E$ and $ F$ are taken respectively on sides $ AB$ and $ AD$ so that $ AE \equal{} AF$ and the quadrilateral $ CDFE$ has maximum area. In square units this maximum area is:
$ \textbf{(A)}\ \frac12 \qquad \textbf{(B)}\ \frac {9}{16} \qquad \textbf{(C)}\ \frac{19}{32} \qquad \textbf{(D)}\ \frac {5}{8} \qquad \textbf{(E)}\ \frac23$
1985 AIME Problems, 9
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
2011 China Northern MO, 1
It is known that the general term $\{a_n\}$ of the sequence is $a_n =(\sqrt3 +\sqrt2)^{2n}$ ($n \in N*$), let $b_n= a_n +\frac{1}{a_n}$ .
(1) Find the recurrence relation between $b_{n+2}$, $b_{n+1}$, $b_n$.
(2) Find the unit digit of the integer part of $a_{2011}$.
2006 AMC 12/AHSME, 11
Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$?
$ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$
$\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$
2023 Indonesia TST, C
Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower.
Prove that the team that fi nished fourth won exactly two games.
1999 Harvard-MIT Mathematics Tournament, 3
How many non-empty subsets of $\{1, 2, 3, 4, 5, 6,7,8\}$ have exactly $k$ elements and do not contain the element $k$ for some $k = 1, 2,...,8$.
2020 Thailand Mathematical Olympiad, 7
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
2024 USA TSTST, 8
Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel.
[i]Michael Ren[/i]
2020 ELMO Problems, P5
Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that
[list]
[*]each row contains $n$ distinct consecutive integers in some order,
[*]each column contains $m$ distinct consecutive integers in some order, and
[*]each entry is less than or equal to $s$.
[/list]
[i]Proposed by Ankan Bhattacharya.[/i]
1985 Putnam, B1
Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.
1979 Putnam, B4
(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.$$ Intelligent guessing of the form of a solution may be helpful.
(b) Let $y=f(x)$ be the solution of the [i]nonhomogeneous[/i] differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1.$ Find integers $a,b,c$ such that $(f(-2)-a)(f(2)-b)=c.$
1971 All Soviet Union Mathematical Olympiad, 148
The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers.
2020 Princeton University Math Competition, A4/B6
Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively prime integers $a, b$. Find $a + b$.
1989 Flanders Math Olympiad, 4
Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer.
If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$,
and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$,
then prove that $f(x)=x^n$ for all $x \in D$.
2002 National Olympiad First Round, 15
There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats?
$
\textbf{a)}\ \dfrac{1}{55}
\qquad\textbf{b)}\ \dfrac{1}{50}
\qquad\textbf{c)}\ \dfrac{2}{55}
\qquad\textbf{d)}\ \dfrac{1}{25}
\qquad\textbf{e)}\ \text{None of above}
$
2015 HMNT, 10
Let $N$ be the number of functions $f$ from $\{1, 2, \dots, 101 \} \rightarrow \{1, 2, \dots, 101 \}$ such that $f^{101}(1) = 2.$ Find the remainder when $N$ is divided by $103.$
2022 Saudi Arabia BMO + EGMO TST, 2.3
A rectangle $R$ is partitioned into smaller rectangles whose sides are parallel with the sides of $R$. Let $B$ be the set of all boundary points of all the rectangles in the partition, including the boundary of $R$. Let S be the set of all (closed) segments whose points belong to $B$. Let a maximal segment be a segment in $S$ which is not a proper subset of any other segment in $S$. Let an intersection point be a point in which $4$ rectangles of the partition meet. Let $m$ be the number of maximal segments, $i$ the number of intersection points and $r$ the number of rectangles. Prove that $m + i = r + 3$.