Found problems: 85335
2021 China Team Selection Test, 1
Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$
2017 Bosnia Herzegovina Team Selection Test, 1
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
2021 AMC 10 Fall, 19
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$
$(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$
2005 Uzbekistan National Olympiad, 4
Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.
2018 PUMaC Live Round, Calculus 1
Freddy the king of flavortext has an infinite chest of coins. For each number \(p\) in the interval \([0, 1]\), Freddy has a coin that has probability \(p\) of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is \(\frac{p}{q}\) for two integers \(p, q\), find \(p + q\).
Note: flavortext is made up
1992 IMO Longlists, 48
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity
\[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\]
Where $\alpha,\beta$ are given real numbers.
2016 Romanian Master of Mathematics Shortlist, C2
A frog trainer places one frog at each vertex of an equilateral triangle $ABC$ of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let $M$ and $N$ be two points on the rays $AB$ and $AC$, respectively, emanating from $A$, such that $AM = AN = \ell$, where $\ell$ is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle $AMN$ (inside or on the boundary), and no more jumps are performed.
Determine the number of final positions the three frogs may reach in the triangle $AMN$. (During the process, the frogs may leave the triangle $AMN$, only their nal positions are to be in that triangle.)
2007 May Olympiad, 4
A $7\times 7$ board has a lamp on each of its $49$ squares, which can be on or off.
The allowed operation is to choose $3$ consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely
[img]https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png[/img]
Give a configuration of exactly $8$ lit lamps located in the first $4$ rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.
2021 China Second Round Olympiad, Problem 7
For two sets $A, B$, define the operation $$A \otimes B = \{x \mid x=ab+a+b, a \in A, b \in B\}.$$ Set $A=\{0, 2, 4, \cdots, 18\}$ and $B=\{98, 99, 100\}$. Compute the sum of all the elements in $A \otimes B$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 7)[/i]
1999 Mongolian Mathematical Olympiad, Problem 5
Find the number of polynomials $P(x)$ of degree $6$ whose coefficients are in the set $\{1,2,\ldots,1999\}$ and which are divisible by $x^3+x^2+x+1$.
2009 Purple Comet Problems, 2
Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime.
2011 IFYM, Sozopol, 2
On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.
2017 Taiwan TST Round 2, 2
Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $
[i]Proposed by Telv Cohl[/i]
2016 CHMMC (Fall), 6
For any nonempty set of integers $X$, define the function $$f(X) = \frac{(-1)^{|X|}}{ \left(\prod_{k\in X} k \right)^2}$$ where $|X|$ denotes the number of elements in $X$.
Consider the set $S = \{2, 3, . . . , 13\}$ . Note that $1$ is not an element of $S$.
Compute $$\sum_{T\subseteq S, T \ne \emptyset} f(T).$$
where the sum is taken over all nonempty subsets $T$ of $S$.
2013 Bosnia and Herzegovina Junior BMO TST, 2
Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality:
$\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$
1967 IMO Shortlist, 2
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
1994 Poland - Second Round, 5
The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$
.
1993 Swedish Mathematical Competition, 1
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.
2021 China Girls Math Olympiad, 3
Find the smallest positive integer $n$, such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied:
(1) the number of cells colored in each color is the same;
(2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell;
(3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.
2007 Harvard-MIT Mathematics Tournament, 1
Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
2017 CHMMC (Fall), 5
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.
2025 VJIMC, 3
Evaluate the integral $\int_0^{\infty} \frac{\log(x+2)}{x^2+3x+2}\mathrm{d}x$.
2023 Austrian MO National Competition, 1
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.
1998 Croatia National Olympiad, Problem 2
Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that
(a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and
(a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.
2018 European Mathematical Cup, 1
Let $a, b, c$ be non-zero real numbers such that $a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}.$ Prove that at least two of $a, b, c$ are equal.