Found problems: 85335
2017 Princeton University Math Competition, B2
Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?
2019 CMIMC, 6
Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$
2005 MOP Homework, 6
A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.
1970 Poland - Second Round, 1
Prove that $$ |\cos n\beta - \cos n\alpha| \leq n^2 |\cos \beta - \cos\alpha|,$$ where $n$ is a natural number . Check for what values of $ n $, $ \alpha $, $ \beta $ equality holds.
2012 Online Math Open Problems, 21
If \[2011^{2011^{2012}} = x^x\] for some positive integer $x$, how many positive integer factors does $x$ have?
[i]Author: Alex Zhu[/i]
2004 AMC 10, 20
In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);
label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$
2017 Iran MO (3rd round), 2
Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that
$$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$
2012 China Girls Math Olympiad, 8
Find the number of integers $k$ in the set $\{0, 1, 2, \dots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.
2015 EGMO, 4
Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers
which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.
2019 Middle European Mathematical Olympiad, 7
Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$. Prove that $c(a-1)+b$ does not divide $c(b-1)+a$.
[i]Proposed by Dominik Burek, Poland[/i]
2008 Grigore Moisil Intercounty, 2
Determine the natural numbers a, b, c s.t. :
$ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $
The challenge here is to come up with as basic solution as possible.
1993 India National Olympiad, 2
Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.
2021 Middle European Mathematical Olympiad, 1
Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying
\[ x_{n+1}=A-\frac{1}{x_n} \]
for every integer $n \ge 1$, has only finitely many negative terms.
1981 All Soviet Union Mathematical Olympiad, 323
The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?
1984 Putnam, A1
Let $A$ be a solid $a\times b\times c$ rectangular brick, where $a,b,c>0$. Let $B$ be the set of all points which are a distance of at most one from some point of $A$. Express the volume of $B$ as a polynomial in $a,b,$ and $c$.
2018 Iran MO (1st Round), 14
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}
2021 JHMT HS, 10
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of
\[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \]
can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$
1988 IMO Longlists, 5
Let $k$ be a positive integer and $M_k$ the set of all the integers that are between $2 \cdot k^2 + k$ and $2 \cdot k^2 + 3 \cdot k,$ both included. Is it possible to partition $M_k$ into 2 subsets $A$ and $B$ such that
\[ \sum_{x \in A} x^2 = \sum_{x \in B} x^2. \]
1995 Poland - Second Round, 3
Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$.
Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.
1973 Spain Mathematical Olympiad, 2
Determine all solutions of the system
$$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$
in which the first three are equations and the last one is a linear inequality.
2023-IMOC, N5
Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2001 District Olympiad, 2
Consider the number $n=123456789101112\ldots 99100101$.
a)Find the first three digits of the number $\sqrt{n}$.
b)Compute the sum of the digits of $n$.
c)Prove that $\sqrt{n}$ isn't rational.
[i]Valer Pop[/i]
2007 Indonesia MO, 4
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called [i]beautiful[/i] if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.
2009 India Regional Mathematical Olympiad, 2
Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.