Found problems: 85335
2022 Moscow Mathematical Olympiad, 1
$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$.
Prove $bc \geq b+c$
2008 Brazil Team Selection Test, 1
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.
[i]Author: Dan Brown, Canada[/i]
2010 LMT, 36
Write down one of the following integers: $1, 2, 4, 8, 16.$ If your team is the only one
that submits this integer, you will receive that number of points; otherwise, you receive zero.
[b][color=#f00]There's no real way to solve this but during the competition, each of the 5 available scores were submitted at least twice by the 16 teams competing. [/color][/b]
2018 Adygea Teachers' Geometry Olympiad, 3
Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.
1935 Moscow Mathematical Olympiad, 010
Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\
x + y + 2z = 4(a^2 + 1) \\
z^2 - xy = a^2
\end{cases}$
2012 Israel National Olympiad, 4
We are given a 7x7 square board. In each square, one of the diagonals is traced, and then one of the two formed triangles is colored blue. What is the largest area a continuous blue component can have?
(Note: continuous blue component means a set of blue triangles connected via their edges, passing through corners is not permitted)
2010 Princeton University Math Competition, 2
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?
2013 BMT Spring, 5
Two positive integers $m$ and $n$ satisfy
$$max \,(m, n) = (m - n)^2$$
$$gcd \,(m, n) = \frac{min \,(m, n)}{6}$$
Find $lcm\,(m, n)$
2012 Purple Comet Problems, 3
The diagram below shows a large square divided into nine congruent smaller squares. There are circles inscribed in five of the smaller squares. The total area covered by all the five circles is $20\pi$. Find the area of the large square.
[asy]
size(80);
defaultpen(linewidth(0.6));
pair cent[] = {(0,0),(0,2),(1,1),(2,0),(2,2)};
for(int i=0;i<=3;++i)
{
draw((0,i)--(3,i));
}
for(int j=0;j<=3;++j)
{
draw((j,0)--(j,3));
}
for(int k=0;k<=4;++k)
{
draw(circle((cent[k].x+.5,cent[k].y+.5),.5));
}
[/asy]
2017 BMT Spring, 5
How many subsets of $\{1, 2,...,9\}$ do not contain $2$ adjacent numbers?
2011 ELMO Shortlist, 4
In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
\[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\]
[i]Calvin Deng.[/i]
2023 SAFEST Olympiad, 1
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$.
[i]Proposed by Liam Baker, South Africa[/i]
2006 Cezar Ivănescu, 2
[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $
[b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that
$$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$
for any nonnegative integer $ n. $
Russian TST 2018, P1
Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.
2009 Today's Calculation Of Integral, 437
Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$
Kyiv City MO Seniors 2003+ geometry, 2004.11.4
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be the feet of the perpendiculars drawn from point $A$ on the lines $A_1D$ and $A_1C$, respectively, and the points $P$ and $Q$ be the feet of the perpendiculars drawn from point $B_1$ on the lines $A_1C_1$ and $A_1C$, respectively. Prove that $\angle EFA = \angle PQB_1$
2019 Saint Petersburg Mathematical Olympiad, 1
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.
2012 Greece Team Selection Test, 1
Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.
2012-2013 SDML (Middle School), 2
When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number?
2015 China Team Selection Test, 5
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
2013 India Regional Mathematical Olympiad, 3
Consider the expression \[2013^2+2014^2+2015^2+ \cdots+n^2\]
Prove that there exists a natural number $n > 2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal $9999$
PEN E Problems, 19
Let $p$ be an odd prime. Without using Dirichlet's theorem, show that there are infinitely many primes of the form $2pk+1$.
2001 Moldova National Olympiad, Problem 2
Let $S(n)$ denote the sum of digits of a natural number $n$. Find all $n$ for which $n+S(n)=2004$.
2007 IMAC Arhimede, 6
Let $A_1A_2...A_n$ ba a polygon. Prove that there is a convex polygon $B_1B_2...B_n$ such that $B_iB_{i + 1} = A_iA_{i + 1}$ for $i \in \{1, 2,...,n-1\}$ and $B_nB_1 = A_nA_1$ (some of the successive vertices of the polygon $B_1B_2...B_n$ can be colinear).
1977 Swedish Mathematical Competition, 5
The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.