Found problems: 85335
2015 Danube Mathematical Competition, 3
Solve in N $a^2 = 2^b3^c + 1$.
2020 MBMT, 26
Let $\triangle MBT$ be a triangle with $\overline{MB} = 4$ and $\overline{MT} = 7$. Furthermore, let circle $\omega$ be a circle with center $O$ which is tangent to $\overline{MB}$ at $B$ and $\overline{MT}$ at some point on segment $\overline{MT}$. Given $\overline{OM} = 6$ and $\omega$ intersects $ \overline{BT}$ at $I \neq B$, find the length of $\overline{TI}$.
[i]Proposed by Chad Yu[/i]
2016 IFYM, Sozopol, 5
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.
2011 Morocco National Olympiad, 2
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
1964 Spain Mathematical Olympiad, 2
The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that:
a) $f(x)$ is a continuous function
b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14.
c) $f(0)=0$ and $f(140000)=14000$.
Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.
2020 USA TSTST, 4
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
2000 All-Russian Olympiad Regional Round, 11.8
There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $N + 2$ republics so that no two cities from the same republic are connected by a road.
2018 Balkan MO Shortlist, N5
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$
Prove that $x=1$.
[i](Silouanos Brazitikos, Greece)[/i]
2011 China Western Mathematical Olympiad, 3
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
2016 Taiwan TST Round 3, 4
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2016 PUMaC Number Theory B, 4
For a positive integer $n$, let $P(n)$ be the product of the factors of $n$ (including $n$ itself).
A positive integer $n$ is called [i]deplorable [/i] if $n > 1$ and $\log_n P(n)$ is an odd integer.
How many factors of $2016$ are [i]deplorable[/i]?
1998 Tuymaada Olympiad, 7
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
1968 Miklós Schweitzer, 4
Let $ f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $ N_k$ of natural numbers such that \[ f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\
.\] Prove that $ f$ is identically $ 1$.
[i]I. Katai[/i]
2006 MOP Homework, 2
Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$.
2018 Balkan MO Shortlist, C2
Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins.
Determine all pairs $(a,b)$ of positive integers such that if initially the two piles have $a$ and $b$ coins respectively, then Bob has a winning strategy.
Proposed by Dimitris Christophides, Cyprus
2011 ISI B.Stat Entrance Exam, 8
Let
\[I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} \, dx , \ \ \ \ n=1,2,3,4\]
Arrange $I_1, I_2, I_3, I_4$ in increasing order of magnitude. Justify your answer.
1970 IMO Longlists, 43
Prove that the equation
\[x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0\]
has one root $x_1 = \tan \frac{\pi}{36}$, and find the other roots.
2016 LMT, 23
Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that
[list]
[*] $a_1+a_2=b_1+b_2,$
[*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and
[*] $a_{n-1}+a_n=b_{n-1}+b_n$.
[/list]
Find the number of junk positive integers less than or equal to $2016$.
[i]Proposed by Nathan Ramesh
2008 All-Russian Olympiad, 5
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells?
2022 Romania EGMO TST, P2
At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.
2014 Greece Junior Math Olympiad, 1
Let $ABC$ be a triangle and let $M$ be the midpoint $BC$. On the exterior of the triangle, consider the parallelogram $BCDE$ such that $BE//AM$ and $BE=AM/2$ . Prove that line $EM$ passes through the midpoint of segment $AD$.
2021 Harvard-MIT Mathematics Tournament., 10
Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white,
such that the following conditions hold:
$\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any $2\times 2$ subgrid contains at least one square of each color.
Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).
2007 Harvard-MIT Mathematics Tournament, 19
Define $x\star y=\frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4}$. Compute \[((\cdots ((2007\star 2006)\star 2005)\star\cdots )\star 1).\]
2010 Peru Iberoamerican Team Selection Test, P5
The trapeze $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle $\Gamma$. Let $X$ be a variable point of the arc $\overarc{AB}$ that does not contain either $C$ or $D$. Let $Y$ be the point of intersection of $AB$ and $DX$, and let $Z$ be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$. Prove that the measure of the angle $\angle AZX$ does not depend on the choice of $X$.
2012 Online Math Open Problems, 44
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
[i]Author: Anderson Wang[/i]