Found problems: 85335
2012 Belarus Team Selection Test, 1
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
2007 Indonesia TST, 2
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.
2016 Postal Coaching, 5
Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.
2013 Waseda University Entrance Examination, 4
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $S(x)$ at the hight $x$.
(2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$
2000 Tournament Of Towns, 2
Each of a pair of opposite faces of a unit cube is marked by a dot. Each of another pair of opposite faces is marked by two dots. Each of the remaining two faces is marked by three dots. Eight such cubes are used to construct a $2\times 2 \times 2$ cube. Count the total number of dots on each of its six faces. Can we obtain six consecutive numbers?
(A Shapovalov)
2010 Middle European Mathematical Olympiad, 5
Three strictly increasing sequences
\[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\]
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:
(a) $c_{a_n}=b_n+1$;
(b) $a_{n+1}>b_n$;
(c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even.
Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 1)[/i]
2015 Azerbaijan JBMO TST, 2
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]
1999 Mongolian Mathematical Olympiad, Problem 1
The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.
2021 Philippine MO, 1
In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.
2022 China Second Round, 2
Integer $n$ has $k$ different prime factors. Prove that $\sigma (n) \mid (2n-k)!$
2024 USA IMO Team Selection Test, 5
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
[i]Ray Li[/i]
1988 IMO Shortlist, 11
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?
1994 AMC 8, 24
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares.
$\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 16$
2016 Korea Winter Program Practice Test, 3
$p, q, r$ are natural numbers greater than 1.
There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions.
(1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$.
(2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.)
If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.
1987 China National Olympiad, 6
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.
V Soros Olympiad 1998 - 99 (Russia), 11.2
Five edges of a triangular pyramid are equal to $1$. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to $1$.
2019 IMAR Test, 4
Show that the length of a cycle that contains every edge of a connected graph is at most the sum between the vertices and nodes of the graph, minus $ 1. $
2021 Irish Math Olympiad, 5
The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$.
You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$.
(a) Find $g(2021)$.
(b) Find $g(1/2021)$.
2020 CCA Math Bonanza, I2
Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$, the distance between their centers are $5$, and the area inside of $\gamma$ but outside of $\omega$ is $100\pi$. What is the sum of the radii of the circles?
[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0), B=(r1-r2,0);
draw(A--B,dashed);
draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));
filldraw(circle(A,dr)); filldraw(circle(B,dr));
label("$5$",(A+B)/2,dir(-90));
label("$\gamma$",A+r1*dir(135),dir(135)); label("$\omega$",B+r2*dir(135),dir(135));
[/asy]
[i]2020 CCA Math Bonanza Individual Round #2[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 9.7
In the acute-angled triangle $ABC$, the points $P$, $N$, $ M$ are the feet of the altitudes drawn from the vertices $C$, $A$, $B$, respectively. The lengths of the projections of the sides $AB$, $BC$, $CA$ on straight lines $MN$, $PM$, $NP$ respectively, are equal to each other. Prove that triangle $ABC$ is regular.
2018 CMIMC Number Theory, 1
Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$?
2011 Purple Comet Problems, 27
Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]
2019 Junior Balkan Team Selection Tests - Moldova, 7
Point $H$ is the orthocenter of the acute triangle $\Delta ABC$ and point $K$,situated on the line $(BC)$, is the foot of the perpendicular from point $A$ .The circle $\Omega$ passes through points $A$ and $K$ ,intersecting the sides $(AB)$ and $(AC)$ in points $M$ and $N$ .The line that passes through point $A$ and is parallel with $BC$ intersects again the circumcircles of triangles $\Delta AHM$ and $\Delta AHN$ in points $X$ and $Y$.Prove that $XY =BC$.
2015 BMT Spring, Tie 3
A bag contains $12$ marbles: $3$ red, $4$ green, and $5$ blue. Repeatedly draw marbles with replacement until you draw two marbles of the same color in a row. What is the expected number of times that you will draw a marble?
1959 Poland - Second Round, 3
Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then
$$ tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.$$