Found problems: 85335
2021 MOAA, 22
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
[i]Proposed by Andy Xu[/i]
2012 Centers of Excellency of Suceava, 3
Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries.
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
2012 JBMO ShortLists, 4
Solve the following equation for $x , y , z \in \mathbb{N}$ :
\[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
2016 Germany Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2011 JBMO Shortlist, 3
Find all positive integers $n$ such that the equation $y^2 + xy + 3x = n(x^2 + xy + 3y)$ has at least a solution $(x, y)$ in positive integers.
1993 India National Olympiad, 5
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
2024 CIIM, 5
A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Prove that
\[
\lim_{N \to \infty} \frac{a(N)}{N}
\]
exists and determine its value.
2002 Flanders Junior Olympiad, 4
Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?
[img]https://cdn.artofproblemsolving.com/attachments/a/8/807bb5b760caaa600f0bac95358963a902b1e7.png[/img]
LMT Speed Rounds, 2010.8
How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$
2017 Saint Petersburg Mathematical Olympiad, 4
A positive integer $n$ is called almost-square if $n$ can be represented as $n=ab$ where $a,b$ are positive integers that $a\leq b\leq 1.01a$. Prove that there exists infinitely many positive integers $m$ that there’re no almost-square positive integer among $m,m+1,…,m+198$.
2005 Junior Balkan Team Selection Tests - Moldova, 4
Let the $A$ be the set of all nonenagative integers.
It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds:
[b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$;
[b]2)[/b] $f(2n) < 6f(n)$,
Find all solutions of $f(k)+f(l) = 293$, $k<l$.
2012 Uzbekistan National Olympiad, 3
The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.
2018 Harvard-MIT Mathematics Tournament, 4
Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.
2005 Gheorghe Vranceanu, 1
Let be a natural number $ n\ge 2 $ and the $ n\times n $ matrix whose entries at the $ \text{i-th} $ line and $ \text{j-th} $ column is $ \min (i,j) . $ Calculate:
[b]a)[/b] its determinant.
[b]b)[/b] its inverse.
2007 Nicolae Păun, 4
Prove that for any natural number $ n, $ there exists a number having $ n+1 $ decimal digits, namely, $ k_0,k_1,k_2,\ldots ,k_n $, and a $ \text{(n+1)-tuple}, $ say $\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , $ that satisfies:
$$ 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2} $$
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
2014 Singapore Junior Math Olympiad, 5
In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.
2021 Turkey Team Selection Test, 1
Let \(n\) be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.
2014 National Olympiad First Round, 13
Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{ADB} \right)=15^{\circ}$, $m \left (\widehat{BCD} \right)=90^{\circ}$. The diagonals of quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC|=4, |EA|=8$ and $|EP|=2$. What is $m \left (\widehat{PBD} \right)$?
$
\textbf{(A)}\ 15^{\circ}
\qquad\textbf{(B)}\ 30^{\circ}
\qquad\textbf{(C)}\ 45^{\circ}
\qquad\textbf{(D)}\ 60^{\circ}
\qquad\textbf{(E)}\ 75^{\circ}
$
1992 Putnam, A3
Let $m,n$ are natural numbers such that $GCD(m,n)=1$.Find all triplets $(x,y,n)$ which sastify $(x^2+y^2)^m=(xy)^n$
2007 Pre-Preparation Course Examination, 2
a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere.
b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.
2008 Harvard-MIT Mathematics Tournament, 9
Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.
2009 China Team Selection Test, 6
Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.
2023 BMT, 7
For an integer $n > 0$, let $p(n)$ be the product of the digits of $n$. Compute the sum of all integers $n$ such that $n - p(n) = 52$.
2016 Junior Regional Olympiad - FBH, 4
In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$
1997 Mexico National Olympiad, 2
In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.