Found problems: 85335
2013 China Western Mathematical Olympiad, 3
Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$
1979 Chisinau City MO, 181
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
1997 VJIMC, Problem 1
Let $a$ be an odd positive integer. Prove that if $d$ divides $a^2+2$, then $d\equiv1\pmod8$ or $d\equiv3\pmod8$.
1999 AMC 12/AHSME, 26
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have?
$ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$
2020 Azerbaijan Senior NMO, 1
$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$
1956 AMC 12/AHSME, 26
Which one of the following combinations of given parts does not determine the indicated triangle?
$ \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle}$
$ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle}$
$ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle}$
$ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle}$
$ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle}$
2015 JHMT, 1
Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius $3$ inches with a sector of $120^o$ cut out. What is the perimeter of the Pacman sticker in inches?
2009 All-Russian Olympiad, 5
Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.
2021 German National Olympiad, 6
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.
II Soros Olympiad 1995 - 96 (Russia), 9.2
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$
2012 IFYM, Sozopol, 1
Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi $, and radius of its circumscribed circle $R=1$.
2008 District Olympiad, 3
Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.
2018 All-Russian Olympiad, 1
The polynomial $P (x)$ is such that the polynomials $P (P (x))$ and $P (P (P (x)))$ are strictly monotone on the whole real axis. Prove that $P (x)$ is also strictly monotone on the whole real axis.
LMT Team Rounds 2010-20, B4
Find the greatest prime factor of $20!+20!+21!$.
1997 Baltic Way, 8
If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.
Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?
2021 Durer Math Competition Finals, 8
Benedek wrote the following $300 $ statements on a piece of paper.
$2 | 1!$
$3 | 1! \,\,\, 3 | 2!$
$4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$
$5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$
$...$
$24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$
$25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$
How many true statements did Benedek write down?
The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.
1997 Iran MO (3rd Round), 5
In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.
2015 Bangladesh Mathematical Olympiad, 5
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
2018 Moldova EGMO TST, 5
Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the inequality
$$\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.$$
[i]Proposed by Baasanjav Battsengel[/i]
2019 Math Prize for Girls Problems, 14
Devah draws a row of 1000 equally spaced dots on a sheet of paper. She goes through the dots from left to right, one by one, checking if the midpoint between the current dot and some remaining dot to its left is also a remaining dot. If so, she erases the current dot. How many dots does Devah end up erasing?
2014 Singapore Senior Math Olympiad, 11
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$
KoMaL A Problems 2018/2019, A. 745
A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.
2006 Purple Comet Problems, 11
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.
Gheorghe Țițeica 2025, P1
Let there be $2n+1$ distinct points on a circle. Consider the set of distances between any two out of the $2n+1$ points. What is the smallest size of this set?
[i]Radu Bumbăcea[/i]
2021 Iranian Geometry Olympiad, 2
Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$.
[i]Proposed by Josef Tkadlec - Czech Republic[/i]