Found problems: 85335
2022 VN Math Olympiad For High School Students, Problem 5
Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.
2023 Kazakhstan National Olympiad, 5
Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.
1966 IMO Longlists, 58
In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
2001 Argentina National Olympiad, 1
Sergio thinks of a positive integer $S$, less than or equal to $100$. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers $A$ and $B$ less than $100$, and asks Sergio what is the greatest common factor between $A+ S$ and $B$. Give a sequence of seven steps that ensures Iván guesses the number $S$ that Sergio thought of.
Clarification:In each step, Sergio correctly answers Iván's question.
2022 IFYM, Sozopol, 4
a) Prove that for each positive integer $n$ the number or ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=n$ is finite and is multiple of 6.
b) Find all ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=727$.
2020 LMT Fall, B7
Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$
Kyiv City MO 1984-93 - geometry, 1991.10.2
In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$, the points $C_1$, $A_1$, and $B_1$ are marked such that the segments $AA_1$, $BB_1$, $CC_1$ intersect at some point $O$ and the angles $AA_1C$, $BB_1A$, $CC_1B$ are equal. Prove that $AA_1$, $BB_1$, and $CC_1$ are the altitudes of the triangle.
2017 European Mathematical Cup, 1
Solve in integers the equation :
$x^2y+y^2=x^3$
1999 Harvard-MIT Mathematics Tournament, 2
A ladder is leaning against a house with its lower end $15$ feet from the house. When the lower end is pulled $9$ feet farther from the house, the upper end slides $13$ feet down. How long is the ladder (in feet)?
2006 IMO Shortlist, 1
We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on.
Initially all the lamps are off except the leftmost one which is on.
$ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off.
$ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.
2004 Korea - Final Round, 1
An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?
2019 Saudi Arabia BMO TST, 1
There are $n$ people with hats present at a party. Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment. Find all $n \ge 2$ for which the order of
greetings can be arranged in such a way that after all of them, each person has their own hat back.
2011 Tournament of Towns, 2
$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$.
[hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]
2017 HMNT, 4
Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?
2021 Malaysia IMONST 1, 7
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
1980 Canada National Olympiad, 1
If $a679b$ is the decimal expansion of a number in base $10$, such that it is divisible by $72$, determine $a,b$.
2014 Contests, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
1996 China Team Selection Test, 3
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
1952 Miklós Schweitzer, 2
Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?
1980 AMC 12/AHSME, 28
The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals
$\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$
2018 BMT Spring, 3
If $A$ is the area of a triangle with perimeter $ 1$, what is the largest possible value of $A^2$?
2020 Harvard-MIT Mathematics Tournament, 10
Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$.
[i]Proposed by James Lin.[/i]
2010 National Chemistry Olympiad, 9
How many neutrons are in $0.025$ mol of the isotope ${ }_{24}^{54}\text{Cr}$?
$ \textbf{(A)}\hspace{.05in}1.5\times10^{22} \qquad\textbf{(B)}\hspace{.05in}3.6\times10^{23} \qquad\textbf{(C)}\hspace{.05in}4.5\times10^{23} \qquad\textbf{(D)}\hspace{.05in}8.1\times10^{23} \qquad $
2022 CCA Math Bonanza, L3.2
In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$.
[i]2022 CCA Math Bonanza Lightning Round 3.2[/i]