Found problems: 85335
PEN E Problems, 3
Find the sum of all distinct positive divisors of the number $104060401$.
2014 Greece Team Selection Test, 3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
KoMaL A Problems 2023/2024, A. 873
Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A', B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$.
[i]Proposed by Áron Bán-Szabó, Budapest[/i]
MBMT Guts Rounds, 2015.26
Choose a real number between $0$ and $10$, inclusive. If your number is less than the average of all numbers chosen, you will get your number's worth of points, but if your number is greater than or equal to the average, you will get $0$ points. For example, if the average of all numbers chosen is $1.2$, and you pick $1.6$, then you will receive $0$ points, but if you pick $0.5$, then you will receive $0.5$ points. Express your answer to the nearest thousandth. For example, $7.800$, $2.110$, and $0.234$ are valid responses, but $7.8$ and $0.2345$ are not. An invalid response will receive a score of zero.
Swiss NMO - geometry, 2019.7
Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.
2019 Korea Junior Math Olympiad., 1
Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.
2010 AMC 10, 13
What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?
$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
2012 Korea Junior Math Olympiad, 3
Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$
2021 BMT, 21
There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.
2013 Serbia Additional Team Selection Test, 3
Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define
$S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ .
$(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence
$S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms?
$(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such
that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms.
[i]Proposed by Milan Basic and Milos Milosavljevic[/i]
2013 National Olympiad First Round, 30
For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
$
\textbf{(A)}\ 212
\qquad\textbf{(B)}\ 206
\qquad\textbf{(C)}\ 191
\qquad\textbf{(D)}\ 185
\qquad\textbf{(E)}\ 173
$
2015 Princeton University Math Competition, B4
A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?
2023 VIASM Summer Challenge, Problem 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$
Prove that: the circumcircle of triangle $MNP$ passes through $O.$
[i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]
2020 Thailand Mathematical Olympiad, 7
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
2019 BMT Spring, 1
A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.
2011 All-Russian Olympiad Regional Round, 9.3
A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the $8\times 8$ chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)
2010 Today's Calculation Of Integral, 623
Find the continuous function satisfying the following equation.
\[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\]
[i]1978 Shibaura Institute of Technology entrance exam[/i]
2014 District Olympiad, 2
Let $ABC$ be a triangle and let the points $D\in BC, E\in AC, F\in AB$, such that \[ \frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB} \]
The half-lines $AD, BE,$ and $CF$ intersect the circumcircle of $ABC$ at points $M,N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.
2009 Tournament Of Towns, 5
Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$
[i](8 points)[/i]
2000 Harvard-MIT Mathematics Tournament, 6
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
2014 Thailand Mathematical Olympiad, 7
Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.
1989 Irish Math Olympiad, 4
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.
1974 Spain Mathematical Olympiad, 6
Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length.
a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$.
b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2005 Sharygin Geometry Olympiad, 17
A circle is inscribed in the triangle $ ABC$ and it's center $I$ and the points of tangency $P, Q, R$ with the sides $BC$, $C A$ and $AB$ are marked, respectively. With a single ruler, build a point $K$ at which the circle passing through the vertices B and $C$ touches (internally) the inscribed circle.