Found problems: 85335
2001 Estonia National Olympiad, 4
We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.
1989 Putnam, A1
How many base ten integers of the form 1010101...101 are prime?
2023 Malaysian Squad Selection Test, 7
Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$.
[i]Proposed by Ivan Chan Kai Chin[/i]
2011 Paraguay Mathematical Olympiad, 5
In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.
2016 Danube Mathematical Olympiad, 4
4.Prove that there exist only finitely many positive integers n such that
$(\frac{n}{1}+1)(\frac{n}{2}+2)...(\frac{n}{n}+n)$ is an integer.
2011 Today's Calculation Of Integral, 690
Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.
1965 AMC 12/AHSME, 22
If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds:
$ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$
$ \textbf{(B)}\ \text{for all values of }x$
$ \textbf{(C)}\ \text{only when }x \equal{} 0$
$ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$
$ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$
Brazil L2 Finals (OBM) - geometry, 2020.1
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
1977 IMO, 3
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)
2007 Ukraine Team Selection Test, 1
$\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)$ such that $a^{2}+b^{2}+c^{2}=1.$ Prove that
$\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).$
2017 Mathematical Talent Reward Programme, MCQ: P 2
$\lim \limits_{x\to \infty} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}=$
[list=1]
[*] $\sqrt{e}$
[*] $\infty$
[*] Does not exists
[*] None of these
[/list]
2021-IMOC, N2
Show that for any two distinct odd primes $p, q$, there exists a positive integer $n$ such that $$\{d(n), d(n + 2) \} = \{p, q\}$$ where $d(n)$ is the smallest prime factor of $n$.
[i]Proposed By - ltf0501[/i]
2001 Stanford Mathematics Tournament, 12
A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.
1981 National High School Mathematics League, 3
Let $\alpha$ be a real number and $\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}$,
$$T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}$$.
$\text{(A)}$$T$ is negative.
$\text{(B)}$$T$ is nonnegative.
$\text{(C)}$$T$ is positive.
$\text{(D)}$$T$ can be either positive or negative.
2014 Iran Geometry Olympiad (senior), 1:
ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.
1980 AMC 12/AHSME, 2
The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is
$\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72$
2023 Girls in Mathematics Tournament, 3
Let $S$ be a set not empty of positive integers and $AB$ a segment with, initially, only points $A$ and $B$ colored by red. An operation consists of choosing two distinct points $X, Y$ colored already by red and $n\in S$ an integer, and painting in red the $n$ points $A_1, A_2,..., A_n$ of segment $XY$ such that $XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY$ and $XA_1<XA_2<...<XA_n$. Find the least positive integer $m$ such exists a subset $S$ of $\{1,2,.., m\}$ such that, after a finite number of operations, we can paint in red the point $K$ in the segment $AB$ defined by $\frac{AK}{KB}= \frac{2709}{2022}$. Also, find the number of such subsets for such a value of $m$.
2001 Baltic Way, 18
Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.
2008 IberoAmerican, 3
Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.
2013 ELMO Shortlist, 3
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
1997 Romania Team Selection Test, 2
Let $a>1$ be a positive integer. Show that the set of integers
\[\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\}\]
contains an infinite subset of pairwise coprime integers.
[i]Mircea Becheanu[/i]
2020 LMT Spring, 24
Let $a$, $b$, and $c$ be real angles such that \newline \[3\sin a + 4\sin b + 5\sin c = 0\] \[3\cos a + 4\cos b + 5\cos c = 0.\] \newline The maximum value of the expression $\frac{\sin b \sin c}{\sin^2 a}$ can be expressed as $\frac{p}{q}$ for relatively prime $p,q$. Compute $p+q$.
2009 Mathcenter Contest, 2
Find all natural numbers that can be written in the form $\frac{4ab}{ab^2+1}$ for some natural $a,b$.
(nooonuii)
2009 Iran MO (3rd Round), 3
3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.
2022 BMT, 23
Carson the farmer has a plot of land full of crops in the shape of a $6 \times 6$ grid of squares. Each day, he uniformly at random chooses a row or a column of the plot that he hasn’t chosen before and harvests all of the remaining crops in the row or column. Compute the expected number of connected components that the remaining crops form after $6$ days. If all crops have been harvested, we say there are $0$ connected components.