Found problems: 85335
2023 Ukraine National Mathematical Olympiad, 10.8
Consider a complete graph on $4046$ nodes, whose edges are colored in some colors. Let's call this graph $k$-good if we can split all its nodes into $2023$ pairs so that there are exactly $k$ distinct colors among the colors of $2023$ edges that connect the nodes from the same pairs. Is it possible that the graph is $999$-good and $1001$-good but not $1000$-good?
[i]Proposed by Anton Trygub[/i]
2008 Harvard-MIT Mathematics Tournament, 4
In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?
2009 ISI B.Stat Entrance Exam, 9
Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of [i]all[/i] points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.
2016 Saudi Arabia BMO TST, 3
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
1952 Miklós Schweitzer, 10
Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$,
$ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$
is positive if $ n$ is odd and negative if $ n$ is even.
1988 Czech And Slovak Olympiad IIIA, 5
Find all numbers $a \in (-2, 2)$ for which the polynomial $x^{154}-ax^{77}+1$ is a multiple of the polynomial $x^{14}-ax^{7}+1$.
2004 Poland - First Round, 3
3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and:
$\frac{EF}{FD}=\frac{AD}{DB}$
Prove that $CF \bot AE$
2009 India Regional Mathematical Olympiad, 3
Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.
1980 IMO, 22
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.
2022-2023 OMMC, 23
Define the Fibonacci numbers such that $F_{1} = F_{2} = 1,$ $F_{k} = F_{k-1} + F_{k-2}$ for $k > 2.$ For large positive integers $n,$ the expression (containing $n$ nested square roots)
$$\sqrt{2023 F^{2}_{2^{1}} + \sqrt{2023 F^{2}_{2^{2}} + \sqrt{2023 F_{2^{3}}^{2} \dots + \sqrt{2023 F^{2}_{2^{n}} }}}}$$
approaches $\frac{a + \sqrt{b}}{c}$ for positive integers $a,b,c$ where $\gcd(a,c) = 1.$ Find $a+b+c.$
2015 Azerbaijan JBMO TST, 2
$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.
2003 Argentina National Olympiad, 2
On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.
2016 CCA Math Bonanza, L2.3
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$
[i]2016 CCA Math Bonanza Lightning #2.3[/i]
2019 LIMIT Category B, Problem 5
A polygon has twice as many diagonals as it has sides. How many sides does it have?
2014 HMNT, 3
The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42,$ and another is a multiple of $72$. What is the minimum possible length of the third side?
2024-IMOC, G2
Triangle $ABC$ has circumcenter $O$. $D$ is an arbitrary point on $BC$, and $AD$ intersects $\odot(ABC)$ at $E$. $S$ is a point on $\odot(ABC)$ such that $D, O, E, S$ are colinear. $AS$ intersects $BC$ at $P$. $Q$ is a point on $BC$ such that $D, O, A, Q$ are concylic. Prove that $\odot(ABC)$ is tangent to $\odot (APQ)$.
[i]Proposed by chengbilly[/i]
2016 PAMO, 4
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.
1990 AMC 12/AHSME, 13
If the following instructions are carried out by a computer, which of $X$ will be printed because of instruction $5$?
$1.$ Start $X$ at $3$ and $S$ at $0$
$2.$ Increase the value of $X$ by $2$.
$3.$ Increase the value of $S$ by the value of $X$.
$4.$ If $S$ is at least $10000$, then go to instsruction $5$; otherwise, go to instruction $2$ and proceed from there.
$5.$ Print the value of $X$.
$6.$ Stop.
$\text{(A)} \ 19 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ 23 \qquad \text{(D)} \ 199 \qquad \text{(E)} \ 201$
2019 Turkey Junior National Olympiad, 1
Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.
2009 India IMO Training Camp, 6
Prove The Following identity:
$ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$.
The Second term on left hand side is to be regarded zero for j=0.
2013 Purple Comet Problems, 10
Find the least positive integer $k$ so that the mean of the numbers $k,k + 1,k + 2,k + 3,\ldots,2k$ exceeds $200$.
2015 Indonesia MO Shortlist, G3
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.
2019 Belarusian National Olympiad, 9.7
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$.
[i](I. Voronovich)[/i]
Champions Tournament Seniors - geometry, 2001.4
Given a convex pentagon $ABCDE$ in which $\angle ABC = \angle AED = 90^o$, $\angle BAC= \angle DAE$. Let $K$ be the midpoint of the side $CD$, and $P$ the intersection point of lines $AD$ and $BK$, $Q$ be the intersection point of lines $AC$ and $EK$. Prove that $BQ = PE$.
2002 Bundeswettbewerb Mathematik, 1
Planet Ypsilon has a calendar similar to ours: A year consists of $365$ days, and every month has $28$, $30$ or $31$ days.
Prove that on Planet Ypsilon, a year must have $12$ months.