Found problems: 85335
May Olympiad L2 - geometry, 2014.2
In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.
2009 Princeton University Math Competition, 2
Find the number of subsets of $\{1,2,\ldots,7\}$ that do not contain two consecutive integers.
2008 Tournament Of Towns, 4
No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.
2010 Saudi Arabia IMO TST, 2
The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.
2021 JHMT HS, 5
Let $\mathcal{S}$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $xy > 0$ and $x^2 + y^2 + 2x + 4y \leq 2021.$ The total area of $\mathcal{S}$ can be written in the form $a\pi + b,$ where $a$ and $b$ are integers. Compute $a + b.$
2022 Greece JBMO TST, 4
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]
1957 AMC 12/AHSME, 23
The graph of $ x^2 \plus{} y \equal{} 10$ and the graph of $ x \plus{} y \equal{} 10$ meet in two points. The distance between these two points is:
$ \textbf{(A)}\ \text{less than 1} \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ \sqrt{2}\qquad
\textbf{(D)}\ 2\qquad
\textbf{(E)}\ \text{more than 2}$
2014 Denmark MO - Mohr Contest, 5
Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.
1997 Italy TST, 2
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
2017 CHMMC (Fall), 4
Let $a = e^{\frac{4\pi i}5}$ be a nonreal fifth root of unity and $b = e^{\frac{2\pi i}{17}}$ be a nonreal seventeenth root of unity. Compute the value of the product \[(a + b) (a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
2013 Brazil Team Selection Test, 2
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
2012 Turkey Junior National Olympiad, 2
In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.
2018 Bangladesh Mathematical Olympiad, 1
Solve:
$x^2(2-x)^2=1+2(1-x)^2$
Where $x$ is real number.
2023 HMNT, 6
There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.
1996 AMC 12/AHSME, 27
Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
balls?
$\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$
2006 Singapore MO Open, 5
Let $a,b,n$ be positive integers. Prove that $n!$ divides \[b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)\]
2013 USAMTS Problems, 3
An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?
2011 Kyrgyzstan National Olympiad, 6
[b]a)[/b] Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times.
[b]b)[/b] Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.
JOM 2015, 3
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.
2009 239 Open Mathematical Olympiad, 1
Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?
2024 AMC 10, 10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$?
$\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$
2011 Greece Team Selection Test, 2
What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping?
Is this problem well-known?
[asy]
size(4.58cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */
/* draw figures */
draw((-3.,2.)--(1.,2.));
draw((-2.,3.)--(-2.,0.));
draw((-2.,0.)--(-1.,0.));
draw((-1.,0.)--(-1.,3.));
draw((-1.,3.)--(-2.,3.));
draw((-3.,1.)--(1.,1.));
draw((1.,1.)--(1.,2.));
draw((-3.,2.)--(-3.,1.));
draw((0.,2.)--(0.,1.));
draw((-1.,2.)--(-1.,1.));
draw((-2.,2.)--(-2.,1.));
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
2011 Greece JBMO TST, 1
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$.
b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$
2005 Purple Comet Problems, 11
The straight river is one and a half kilometers wide and has a current of $8$ kilometers per hour. A boat capable of traveling $10$ kilometers per hour in still water, sets out across the water. How many minutes will it take the boat to reach a point directly across from where it started?
2023 Durer Math Competition Finals, 2
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\
[b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.