Found problems: 85335
2007 Harvard-MIT Mathematics Tournament, 14
We are given some similar triangles. Their areas are $1^2,3^2,5^2,\cdots,$ and $49^2$. If the smallest triangle has a perimeter of $4$, what is the sum of all the triangles' perimeters?
2007 Baltic Way, 4
Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that
\[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]
1958 November Putnam, A1
Let $f(m,1)=f(1,n)=1$ for $m\geq 1, n\geq 1$ and let $f(m,n)=f(m-1, n)+ f(m, n-1) +f(m-1 ,n-1)$ for $m>1$ and $n>1$. Also let
$$ S(n)= \sum_{a+b=n} f(a,b) \,\,\;\; a\geq 1 \,\, \text{and} \,\; b\geq 1.$$
Prove that
$$S(n+2) =S(n) +2S(n+1) \,\, \; \text{for} \, \, n \geq 2.$$
1991 Baltic Way, 18
Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?
2015 Spain Mathematical Olympiad, 1
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?
2019 Kyiv Mathematical Festival, 5
Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019^4?$
2023 All-Russian Olympiad, 5
If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.
1998 IberoAmerican Olympiad For University Students, 6
Take the following differential equation:
\[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\]
If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.
2019 All-Russian Olympiad, 3
We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin?
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
2017 CMIMC Geometry, 1
Let $ABC$ be a triangle with $\angle BAC=117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD\sim\triangle ACB$. Compute the measure of $\angle ABC$, in degrees.
2007 Junior Balkan Team Selection Tests - Romania, 2
Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.
KoMaL A Problems 2019/2020, A. 779
Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$
Prove that lines $PK$ are concurrent.
2018 Romania National Olympiad, 1
Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$
2014 Peru MO (ONEM), 3
a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$.
b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.
1989 Putnam, B3
Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy
$$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define
$$\mu_n=\int^\infty_0x^nf(x)dx.$$
$a.$ Express $\mu_n$ in terms of $\mu_0$.
$b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.
2019 HMNT, 3
For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of cheese on top of the toast that is tangent to each side of the triangle. What is the area of this piece of cheese?
2014 District Olympiad, 3
[list=a]
[*]Let $A$ be a matrix from $\mathcal{M}_{2}(\mathbb{C})$, $A\neq aI_{2}$,
for any $a\in\mathbb{C}$. Prove that the matrix $X$ from $\mathcal{M}
_{2}(\mathbb{C})$ commutes with $A$, that is, $AX=XA$, if and only if there
exist two complex numbers $\alpha$ and $\alpha^{\prime}$, such that $X=\alpha
A+\alpha^{\prime}I_{2}$.
[*]Let $A$, $B$ and $C$ be matrices from $\mathcal{M}_{2}(\mathbb{C})$, such
that $AB\neq BA$, $AC=CA$ and $BC=CB$. Prove that $C$ commutes with all
matrices from $\mathcal{M}_{2}(\mathbb{C})$.[/list]
2004 Junior Tuymaada Olympiad, 4
Given the disjoint finite sets of natural numbers $ A $ and $ B $, consisting of $ n $ and $ m $ elements, respectively. It is known that every natural number belonging to $ A $ or $ B $ satisfies at least one of the conditions $ k + 17 \in A $, $ k-31 \in B $. Prove that $ 17n = 31m $
2015 ASDAN Math Tournament, 1
Four unit circles are placed on a square of side length $2$, with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
MOAA Team Rounds, 2021.16
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees.
[i]Proposed by Raina Yang[/i]
2012 Oral Moscow Geometry Olympiad, 1
In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$.
2024 All-Russian Olympiad, 2
Call a triple $(a,b,c)$ of positive numbers [i]mysterious [/i]if
\[\sqrt{a^2+\frac{1}{a^2c^2}+2ab}+\sqrt{b^2+\frac{1}{b^2a^2}+2bc}+\sqrt{c^2+\frac{1}{c^2b^2}+2ca}=2(a+b+c).\]
Prove that if the triple $(a,b,c)$ is mysterious, then so is the triple $(c,b,a)$.
[i]Proposed by A. Kuznetsov, K. Sukhov[/i]
2024-25 IOQM India, 7
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
LMT Team Rounds 2010-20, A26
Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$.
[i]Proposed by Jeff Lin[/i]