Found problems: 85335
2018 India IMO Training Camp, 2
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
2018 PUMaC Live Round, Estimation 1
A $2$-by-$2018$ grid is completely covered by non-overlapping L-tiles (see diagram below) and $1$-by-$1$ squares. If the L-tiles can be rotated and flipped, there are a total of $M$ such tilings.
[asy]
size(1cm);
draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
draw((0,1)--(1,1)--(1,0));
[/asy]
What is $\ln M?$
Give your answer as an integer or decimal. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor7.5-\tfrac{|A-C|^{1.5}}{20}\rfloor,0\}.$
1995 Hungary-Israel Binational, 2
Let $ P_1$, $ P_2$, $ P_3$, $ P_4$ be five distinct points on a circle. The distance of $ P$ from the line $ P_iP_k$ is denoted by $ d_{ik}$. Prove that $ d_{12}d_{34} \equal{} d_{13}d_{24}$.
2013 Romania Team Selection Test, 4
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
1969 IMO Longlists, 10
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$
2025 AIME, 5
There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Let N be the number of these integers that are divisible by $22$. Find the difference between $N$ and 2025.
2012 Flanders Math Olympiad, 1
Our class decides to have a alpha - beta - gamma tournament. This party game is always played in groups of three. Any possible combination of three players (three students or two students and the teacher) plays the game $1$ time. The player who wins gets $1$ point. The two losers get no points. At the end of the tournament, miraculously, all students have as many points. The teacher has $3$ points. How many students are there in our class?
2007 Peru MO (ONEM), 4
Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.
2010 Contests, 2
The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?
2009 Stanford Mathematics Tournament, 4
Find all values of $x$ for which $f(x)+xf\left(\frac{1}{x}\right)=x$ for any function $f(x)$
2010 LMT, 20
Three vertices of a parallelogram are $(2,-4),(-2,8),$ and $(12,7.)$ Determine the sum of the three possible x-coordinates of the fourth vertex.
1991 IMO, 1
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
2020 Harvard-MIT Mathematics Tournament, 9
Let $P(x)=x^{2020}+x+2$, which has $2020$ distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q(\alpha^2)^2$.
[i]Proposed by Milan Haiman.[/i]
2004 South East Mathematical Olympiad, 5
For $\theta\in[0, \dfrac{\pi}{2}]$, the following inequality $\sqrt{2}(2a+3)\cos(\theta-\dfrac{\pi}{4})+\dfrac{6}{\sin\theta+\cos\theta}-2\sin2\theta<3a+6$ is always true.
Determine the range of $a$.
1991 Arnold's Trivium, 50
Calculate
\[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]
2019 HMNT, 1
Dylan has a $100\times 100$ square, and wants to cut it into pieces of area at least $1$. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?
2021 MOAA, 9
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
[i]Proposed by William Yue[/i]
2024 Iran Team Selection Test, 4
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real numbers $x , y$ this equality holds :
$$f(yf(x)+f(x)f(y))=xf(y)+f(xy)$$
[i]Proposed by Navid Safaei[/i]
1996 Cono Sur Olympiad, 2
Consider a sequence of real numbers defined by:
$a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$
Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.
2023 Romania National Olympiad, 2
We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order).
a) Determine the largest special number $m$ whose sum of digits is equal to $2023$.
b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.
2015 FYROM JBMO Team Selection Test, 3
Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.
2013 Dutch Mathematical Olympiad, 5
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result?
1986 Bulgaria National Olympiad, Problem 6
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.
KoMaL A Problems 2023/2024, A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
2010 Baltic Way, 12
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.
$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.
$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.