Found problems: 85335
1996 Greece Junior Math Olympiad, 1
Solve the equation
$(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$
2021 CHMMC Winter (2021-22), 1
Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$, point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $\overline{OE}$, respectively. Compute the area of quadrilateral $W XY Z$.
1941 Moscow Mathematical Olympiad, 085
Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.
1995 Romania Team Selection Test, 3
Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.
1973 Bundeswettbewerb Mathematik, 4
$n$ persons sit around a round table. The number of persons having the same gender than the person at the right of them is the same as the number of those it isn't true for.
Show that $4|n$.
2020 Ukrainian Geometry Olympiad - April, 3
Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.
2017 District Olympiad, 3
Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers.
[b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $
[b]b)[/b] $ S_5 $ is not a perfect square.
2017 Romania National Olympiad, 1
Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.
2015 Korea National Olympiad, 4
For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$
1971 Miklós Schweitzer, 8
Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.)
[i]A. Adam[/i]
1951 Poland - Second Round, 2
In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$
BD \colon DC = CE \colon EA = AF \colon FB = k,$$
where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $
2009 Harvard-MIT Mathematics Tournament, 6
How many sequences of $5$ positive integers $(a,b,c,d,e)$ satisfy $abcde\leq a+b+c+d+e\leq10$?
2012 Canadian Mathematical Olympiad Qualification Repechage, 5
Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies \[a_{i+1} = a_i +d(a_i),\] for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are infinitely many
terms in the sequence divisible by $3^{2011}$.
2016 AMC 10, 6
Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$
1990 AMC 12/AHSME, 28
A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$.
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16 $
2002 All-Russian Olympiad, 4
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
2010 Kosovo National Mathematical Olympiad, 3
Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).
2018 CCA Math Bonanza, TB3
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$?
[i]2018 CCA Math Bonanza Tiebreaker Round #3[/i]
2024 USAJMO, 1
Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral.
[i]Proposed by Evan O'Dorney[/i]
2006 Korea National Olympiad, 6
Prove that for any positive real numbers $x,y$ and $z,$
$xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$
2013 Princeton University Math Competition, 4
Let $f(x)=1-|x|$. Let \begin{align*}f_n(x)&=(\overbrace{f\circ \cdots\circ f}^{n\text{ copies}})(x)\\g_n(x)&=|n-|x| |\end{align*} Determine the area of the region bounded by the $x$-axis and the graph of the function $\textstyle\sum_{n=1}^{10}f(x)+\textstyle\sum_{n=1}^{10}g(x).$
2012 Math Prize For Girls Problems, 2
In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2.
[asy]
import graph;
pair A = 2dir(0);
pair B = 2dir(60);
pair C = 2dir(120);
pair D = 2dir(180);
pair E = 2dir(240);
pair F = 2dir(300);
path hexagon = A -- B -- C -- D -- E -- F -- cycle;
fill(hexagon, gray);
filldraw(Circle(A, 1), white);
filldraw(Circle(B, 1), white);
filldraw(Circle(C, 1), white);
filldraw(Circle(D, 1), white);
filldraw(Circle(E, 1), white);
filldraw(Circle(F, 1), white);
draw(hexagon);
[/asy]
Adjacent circles are tangent to each other. What is the area of the shaded region?
2015 Flanders Math Olympiad, 2
Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.
2024 Saint Petersburg Mathematical Olympiad, 1
In the cells of the $2024\times 2024$ board, integers are arranged so that in any $2 \times 2023$ rectangle (vertical or horizontal) with one cut corner cell that does not go beyond the board, the sum of the numbers is divided by $13$. Prove that the sum of all the numbers on the board is divisible by $13$.
1999 Swedish Mathematical Competition, 2
Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.