Found problems: 85335
2002 Romania National Olympiad, 2
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that:
$a)$ $f$ is continuous;
$b)$ $f$ is strictly monotone.
2012 AMC 8, 6
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches?
$\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $
1990 Greece Junior Math Olympiad, 2
For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?
1988 IMO Longlists, 13
Let $T$ be a triangle with inscribed circle $C.$ A square with sides of length $a$ is circumscribed about the same circle $C.$ Show that the total length of the parts of the edge of the square interior to the triangle $T$ is at least $2 \cdot a.$
2011 ELMO Shortlist, 1
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.
[i]Evan O'Dorney.[/i]
2010 Junior Balkan Team Selection Tests - Moldova, 8
What is the minimum $n$ so that grid $nxn$ can be covered with equal number of 2x2 squares and angle triminoes (2x2 without one square)
2014 Dutch IMO TST, 1
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.
1984 All Soviet Union Mathematical Olympiad, 379
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
1908 Eotvos Mathematical Competition, 1
Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.
2019 Spain Mathematical Olympiad, 6
In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$.
The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$.
Prove that angles $EDG$ and $ADF$ are equal.
Kyiv City MO Seniors 2003+ geometry, 2012.11.3
Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$.
(Nagel Igor)
2024 Ukraine National Mathematical Olympiad, Problem 4
The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x)\cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.
[i]Remarks.[/i] A linear function $y = kx+b$ is called non-constant if $k\neq 0$.
[i]Proposed by Oleksiy Masalitin[/i]
2010 Contests, 1
Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2013 Princeton University Math Competition, 4
Compute the smallest integer $n\geq 4$ such that $\textstyle\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\textstyle\binom n4$ are $0000$).
2014 Hanoi Open Mathematics Competitions, 5
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
2019 Math Prize for Girls Problems, 7
Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:
I. The quadratic sends integers to integers.
II. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
III. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.
How many quadratics are there with these properties?
2020 Taiwan TST Round 1, 4
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
2011 Saudi Arabia Pre-TST, 2.4
Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.
Kvant 2019, M2555
In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $1000$.
[i]Folklore[/i]
2025 Euler Olympiad, Round 1, 7
Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum:
$$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$
[i]Proposed by Lia Chitishvili, Georgia [/i]
2017 Hanoi Open Mathematics Competitions, 9
Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.
2017 Yasinsky Geometry Olympiad, 2
Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.
2007 Purple Comet Problems, 4
Terry drove along a scenic road using $9$ gallons of gasoline. Then Terry went onto the freeway and used $17$ gallons of gasoline. Assuming that Terry gets $6.5$ miles per gallon better gas mileage on the freeway than on the scenic road, and Terry’s average gas mileage for the entire trip was $30$ miles per gallon, find the number of miles Terry drove.
1986 Federal Competition For Advanced Students, P2, 3
Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by:
$ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$
contains infinitely many natural numbers.
2011 Belarus Team Selection Test, 1
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
I.Voronovich