Found problems: 85335
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]
1998 North Macedonia National Olympiad, 3
A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.
2021 Benelux, 4
A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.
2013 Stanford Mathematics Tournament, 6
A positive integer $b\geq 2$ is [i]neat[/i] if and only if there exist positive base-$b$ digits $x$ and $y$ (that is, $x$ and $y$ are integers, $0<x<b$ and $0<y<b$) such that the number $x.y$ base $b$ (that is, $x+\tfrac yb$) is an integer multiple of $x/y$. Find the number of [i]neat[/i] integers less than or equal to $100$.
2017 Brazil Team Selection Test, 1
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
1990 Tournament Of Towns, (272) 6
A deck consists of $n$ different cards. A move consists of taking out a group of cards in sequence from some place in the deck, and putting it back someplace else without changing the order within the group or turning any cards over. We are required to reverse the order of cards in the deck by such moves.
(a) Prove that for $n = 9$, this can be done in $5$ moves.
(b) Prove that for $n = 52$, this
i. can be done in $27$ moves,
ii. can’t be done in $17$ moves,
iii. can’t be done in $26$ moves.
(SM Voronin, Tchelyabinsk)
2011 ELMO Shortlist, 1
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that
a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;
b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.
Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if
\[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\]
[i]Evan O'Dorney.[/i]
2015 Sharygin Geometry Olympiad, P15
The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.
2015 Miklos Schweitzer, 4
Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.
2005 IberoAmerican Olympiad For University Students, 1
Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find
\[\lim_{n\to\infty}f(n)^{1/n}\]
2018 Oral Moscow Geometry Olympiad, 1
Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second.
[img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]
2021-2022 OMMC, 16
In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$.
[i]Proposed by Vivian Loh [/i]
2009 AMC 8, 14
Austin and Temple are $ 50$ miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging $ 60$ miles per hour. Leaving the car with her daughter, Bonnie rod a bus back to Austin along the same route and averaged $ 40$ miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
$ \textbf{(A)}\ 46 \qquad
\textbf{(B)}\ 48 \qquad
\textbf{(C)}\ 50 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 54$