Found problems: 85335
2023 EGMO, 5
We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its [i]twist[/i] $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$.
Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.
2023 Harvard-MIT Mathematics Tournament, 26
Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$
2014 AMC 8, 9
In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
[asy]
size(300);
defaultpen(linewidth(0.8));
pair A=(-1,0),C=(1,0),B=dir(40),D=origin;
draw(A--B--C--A);
draw(D--B);
dot("$A$", A, SW);
dot("$B$", B, NE);
dot("$C$", C, SE);
dot("$D$", D, S);
label("$70^\circ$",C,2*dir(180-35));
[/asy]
$\textbf{(A) }100\qquad\textbf{(B) }120\qquad\textbf{(C) }135\qquad\textbf{(D) }140\qquad \textbf{(E) }150$
1970 Canada National Olympiad, 9
Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \] a) Give a formula for $f(n)$.
b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.
2003 Junior Balkan Team Selection Tests - Moldova, 5
Prove that each positive integer is equal to a difference of two positive integers with the same number of the prime divisors.
1989 Spain Mathematical Olympiad, 3
Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $
2009 Cuba MO, 5
Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.
2010 Contests, 2
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
2001 AMC 8, 1
Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?
$ \text{(A)}\ 4\qquad\text{(B)}\ 6\qquad\text{(C)}\ 8\qquad\text{(D)}\ 10\qquad\text{(E)}\ 12 $
Russian TST 2015, P1
The points $A', B', C', D'$ are selected respectively on the sides $AB, BC, CD, DA$ of the cyclic quadrilateral $ABCD$. It is known that $AA' = BB' = CC' = DD'$ and \[\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.\]Prove that $ABCD$ is a square.
1999 Harvard-MIT Mathematics Tournament, 6
Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need?
1952 AMC 12/AHSME, 16
If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by:
$ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$
2002 CentroAmerican, 5
Find a set of infinite positive integers $ S$ such that for every $ n\ge 1$ and whichever $ n$ distinct elements $ x_1,x_2,\cdots, x_n$ of S, the number $ x_1\plus{}x_2\plus{}\cdots \plus{}x_n$ is not a perfect square.
1991 Arnold's Trivium, 16
What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?
2019 Math Prize for Girls Problems, 10
A $1 \times 5$ rectangle is split into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?
2015 Princeton University Math Competition, A5
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum
\[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?
2017 Sharygin Geometry Olympiad, 3
$ABCD$ is convex quadrilateral. If $W_a$ is product of power of $A$ about circle $BCD$ and area of triangle $BCD$. And define $W_b,W_c,W_d$ similarly.prove $W_a+W_b+W_c+W_d=0$
2008 JBMO Shortlist, 2
Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infi nite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.
2020 Yasinsky Geometry Olympiad, 2
It is known that the angles of the triangle $ABC$ are $1: 3: 5$. Find the angle between the bisector of the largest angle of the triangle and the line containing the altitude drawn to the smallest side of the triangle.
2016 Harvard-MIT Mathematics Tournament, 8
For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by
\[
W(n,k) =
\begin{cases}
n^n & k = 0 \\
W(W(n,k-1), k-1) & k > 0.
\end{cases}
\]
Find the last three digits in the decimal representation of $W(555,2)$.
2006 Abels Math Contest (Norwegian MO), 3
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.
(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
1952 AMC 12/AHSME, 39
If the perimeter of a rectangle is $ p$ and its diagonal is $ d$, the difference between the length and width of the rectangle is:
$ \textbf{(A)}\ \frac {\sqrt {8d^2 \minus{} p^2}}{2} \qquad\textbf{(B)}\ \frac {\sqrt {8d^2 \plus{} p^2}}{2} \qquad\textbf{(C)}\ \frac {\sqrt {6d^2 \minus{} p^2}}{2}$
$ \textbf{(D)}\ \frac {\sqrt {6d^2 \plus{} p^2}}{2} \qquad\textbf{(E)}\ \frac {8d^2 \minus{} p^2}{4}$
2005 Junior Balkan MO, 1
Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]
2024 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.
2012 Iran MO (3rd Round), 4
Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.