Found problems: 85335
1996 All-Russian Olympiad, 5
Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others?
[i]A. Golovanov[/i]
2001 USA Team Selection Test, 9
Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and
\[\prod_{x\in B} x = \sum_{x\in B} x^2.\]
2014 AMC 8, 4
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$
2019 Baltic Way, 1
For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality
$$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$
2018 MIG, 4
What is the positive difference between the sum of the first $5$ positive even integers and the first $5$ positive odd integers?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
1970 Miklós Schweitzer, 5
Prove that two points in a compact metric space can be joined with a rectifiable arc if and only if there exists a positive number $ K$ such that, for any $ \varepsilon>0$, these points can be connected with an $ \varepsilon$-chain not longer that $ K$.
[i]M. Bognar[/i]
2010 District Olympiad, 1
Prove that any continuos function $ f: \mathbb{R}\rightarrow \mathbb{R}$ with
\[ f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\
a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right.\]
where $ a_1,a_2,b_1,b_2\in \mathbb{R}$, can be written as:
\[ f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R}\]
where $ m_1,m_2,n_1,n_2\in \mathbb{R}$ and $ \epsilon\in \{\minus{}1,\plus{}1\}$.
1975 USAMO, 2
Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]
2010 Bulgaria National Olympiad, 2
Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.
2024 AIME, 15
Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
[asy]
unitsize(40);
real r = pi/6;
pair A1 = (cos(r),sin(r));
pair A2 = (cos(2r),sin(2r));
pair A3 = (cos(3r),sin(3r));
pair A4 = (cos(4r),sin(4r));
pair A5 = (cos(5r),sin(5r));
pair A6 = (cos(6r),sin(6r));
pair A7 = (cos(7r),sin(7r));
pair A8 = (cos(8r),sin(8r));
pair A9 = (cos(9r),sin(9r));
pair A10 = (cos(10r),sin(10r));
pair A11 = (cos(11r),sin(11r));
pair A12 = (cos(12r),sin(12r));
draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle);
filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2));
draw(A4--A11);
draw(0.365*A3--0.365*A12, linewidth(1.2));
dot(A1);
dot(A2);
dot(A3);
dot(A4);
dot(A5);
dot(A6);
dot(A7);
dot(A8);
dot(A9);
dot(A10);
dot(A11);
dot(A12);
[/asy]
2020 Harvard-MIT Mathematics Tournament, 6
Let $ABC$ be a triangle with $AB=5$, $BC=6$, $CA=7$. Let $D$ be a point on ray $AB$ beyond $B$ such that $BD=7$, $E$ be a point on ray $BC$ beyond $C$ such that $CE=5$, and $F$ be a point on ray $CA$ beyond $A$ such that $AF=6$. Compute the area of the circumcircle of $DEF$.
[i]Proposed by James Lin.[/i]
MBMT Guts Rounds, 2015.3
A positive integer $n$ is divisible by $3$ and $5$, but not by $2$. If $n > 20$, what is the smallest possible value of $n$?
1952 AMC 12/AHSME, 32
$ K$ takes $ 30$ minutes less time than $ M$ to travel a distance of $ 30$ miles. $ K$ travels $ \frac {1}{3}$ mile per hour faster than $ M$. If $ x$ is $ K$'s rate of speed in miles per hours, then $ K$'s time for the distance is:
$ \textbf{(A)}\ \dfrac{x \plus{} \frac {1}{3}}{30} \qquad\textbf{(B)}\ \dfrac{x \minus{} \frac {1}{3}}{30} \qquad\textbf{(C)}\ \dfrac{30}{x \plus{} \frac {1}{3}} \qquad\textbf{(D)}\ \frac {30}{x} \qquad\textbf{(E)}\ \frac {x}{30}$
2011 Tournament of Towns, 2
In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.
2005 China Team Selection Test, 2
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
1997 Estonia Team Selection Test, 3
It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.
2015 Harvard-MIT Mathematics Tournament, 7
Suppose $(a_1,a_2,a_3,a_4)$ is a 4-term sequence of real numbers satisfying the following two conditions:
[list]
[*] $a_3=a_2+a_1$ and $a_4=a_3+a_2$;
[*] there exist real numbers $a,b,c$ such that \[an^2+bn+c=\cos(a_n)\] for all $n\in\{1,2,3,4\}$.
[/list]
Compute the maximum possible value of \[\cos(a_1)-\cos(a_4)\] over all such sequences $(a_1,a_2,a_3,a_4)$.
1949-56 Chisinau City MO, 12
Factor the polynomial $bc (b+c) +ca (c-a)-ab(a + b)$.
2014 Sharygin Geometry Olympiad, 8
Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.
(Tran Quang Hung)
1996 Portugal MO, 3
A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?
VI Soros Olympiad 1999 - 2000 (Russia), 8.6
Two players take turns writing down all proper non-decreasing fractions with denominators from $1 $ to $1999$ and at the same time writing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?
2024 Malaysia IMONST 2, 6
There are $2n$ points on a circle, $n$ are red and $n$ are blue. Janson found a red frog and a blue frog at a red point and a blue point on the circle respectively. Every minute, the red frog moves to the next red point in the clockwise direction and the blue frog moves to the next blue point in the anticlockwise direction.
Prove that for any initial position of the two frogs, Janson can draw a line through the circle, such that the two frogs are always on opposite sides of the line.
2009 Indonesia TST, 2
Consider the following array:
\[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots
\] Find the 5-th number on the $ n$-th row with $ n>5$.
2020 LMT Fall, 33
Let $\omega_1$ and $\omega_2$ be two circles that intersect at two points: $A$ and $B$. Let $C$ and $E$ be on $\omega_1$, and $D$ and $F$ be on $\omega_2$ such that $CD$ and $EF$ meet at $B$ and the three lines $CE$, $DF$, and $AB$ concur at a point $P$ that is closer to $B$ than $A$. Let $\Omega$ denote the circumcircle of $\triangle DEF$. Now, let the line through $A$ perpendicular to $AB$ hit $EB$ at $G$, $GD$ hit $\Omega$ at $J$, and $DA$ hit $\Omega$ again at $I$. A point $Q$ on $IE$ satisfies that $CQ=JQ$. If $QJ=36$, $EI=21$, and $CI=16$, then the radius of $\Omega$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a, c) = 1$. Find $a+b+c$.
[i]Proposed by Kevin Zhao[/i]
2019 PUMaC Individual Finals A, B, A1
Given the graph $G$ and cycle $C$ in it, we can perform the following operation: add another vertex $v$ to the graph, connect it to all vertices in $C$ and erase all the edges from $C$. Prove that we cannot perform the operation indefinitely on a given graph.