Found problems: 85335
2017 Bundeswettbewerb Mathematik, 2
In a convex regular $35$-gon $15$ vertices are colored in red. Are there always three red vertices that make an isosceles triangle?
1972 Canada National Olympiad, 8
During a certain election campaign, $p$ different kinds of promises are made by the different political parties ($p>0$). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.
1993 All-Russian Olympiad, 1
For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?
2017 Thailand TSTST, 3
Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.
2012 Today's Calculation Of Integral, 829
Let $a$ be a positive constant. Find the value of $\ln a$ such that
\[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]
2017 Azerbaijan BMO TST, 3
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
2003 JHMMC 8, 4
A number plus $4$ is $2003$. What is the number?
2018 Pan-African Shortlist, C1
A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$.
How many players took part at the tournament?
Russian TST 2019, P2
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2004 IMO Shortlist, 7
For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.
2011 Brazil Team Selection Test, 3
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2024 Vietnam National Olympiad, 6
For each positive integer $n$, let $\tau (n)$ be the number of positive divisors of $n$.
a) Find all positive integers $n$ such that $\tau(n)+2023=n$.
b) Prove that there exist infinitely many positive integers $k$ such that there are exactly two positive integers $n$ satisfying $\tau(kn)+2023=n$.
2008 Moldova National Olympiad, 12.5
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.
2024 Polish MO Finals, 3
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
2021 Estonia Team Selection Test, 2
Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$
2018 BMT Spring, 3
$2018$ people (call them $A, B, C, \ldots$) stand in a line with each permutation equally likely. Given that $A$ stands before $B$, what is the probability that $C$ stands after $B$?
2022 Thailand Online MO, 6
Let $n$ and $k$ be positive integers. Chef Kao cuts a circular pizza through $k$ diameters, dividing the pizza into $2k$ equal pieces. Then, he dresses the pizza with $n$ toppings. For each topping, he chooses $k$ consecutive pieces of pizza and puts that topping on all of the chosen pieces. Then, for each piece of pizza, Chef Kao counts the number of distinct toppings on it, yielding $2k$ numbers. Among these numbers, let $m$ and $M$ being the minimum and maximum, respectively. Prove that $m + M = n$.
MOAA Team Rounds, 2019.2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2024 Harvard-MIT Mathematics Tournament, 16
Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$
1982 IMO Longlists, 23
Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.
2020 LMT Fall, 15
$\triangle ABC$ has $AB=5,BC=6,$ and $AC=7.$ Let $M$ be the midpoint of $BC,$ and let the circumcircle of $\triangle ABM$ intersect $AC$ at $N.$ If the length of segment $MN$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b$ find $a+b.$
[i]Proposed by Alex Li[/i]
2004 Thailand Mathematical Olympiad, 17
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.
2024 AMC 10, 4
Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2014 China National Olympiad, 3
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:
i) $f(1)=f(2)=1$;
ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.
For each integer $m\ge 2$, find the value of $f(2^m)$.
2013 SEEMOUS, Problem 2
Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that
$$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that
$$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$