Found problems: 85335
2004 Swedish Mathematical Competition, 4
If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.
Champions Tournament Seniors - geometry, 2002.2
The point $P$ is outside the circle $\omega$ with center $O$. Lines $\ell_1$ and $\ell_2$ pass through a point $P$, $\ell_1$ touches the circle $\omega$ at the point $A$ and $\ell_2$ intersects $\omega$ at the points $B$ and $C$. Tangent to the circle $\omega$ at points $B$ and $C$ intersect at point $Q$. Let $K$ be the point of intersection of the lines $BC$ and $AQ$. Prove that $(OK) \perp (PQ)$.
2014 Romania National Olympiad, 2
Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $
Show that $ f $ and $ g $ are the same function.
2014 NZMOC Camp Selection Problems, 6
Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.
2012 USAMO, 5
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
2019 Regional Olympiad of Mexico Southeast, 2
Let $ABCD$ a convex quadrilateral. Suppose that the circumference with center $B$ and radius $BC$ is tangent to $AD$ in $F$ and the circumference with center $A$ and radius $AD$ is tangent to $BC$ in $E$. Prove that $DE$ and $CF$ are perpendicular.
2011 Denmark MO - Mohr Contest, 5
Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$.
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1958 AMC 12/AHSME, 30
If $ xy \equal{} b$ and $ \frac{1}{x^2} \plus{} \frac{1}{y^2} \equal{} a$, then $ (x \plus{} y)^2$ equals:
$ \textbf{(A)}\ (a \plus{} 2b)^2\qquad
\textbf{(B)}\ a^2 \plus{} b^2\qquad
\textbf{(C)}\ b(ab \plus{} 2)\qquad
\textbf{(D)}\ ab(b \plus{} 2)\qquad
\textbf{(E)}\ \frac{1}{a} \plus{} 2b$
1987 All Soviet Union Mathematical Olympiad, 461
All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.
1997 Canadian Open Math Challenge, 10
Consider the ten numbers $ar, ar^2, ar^3, ... , ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, determine their product.
2004 Baltic Way, 15
A circle is divided into $13$ segments, numbered consecutively from $1$ to $13$. Five fleas called $A,B,C,D$ and $E$ are sitting in the segments $1,2,3,4$ and $5$. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments $1,2,3,4,5$, but possibly in some other order than they started. Which orders are possible ?
2015 Czech and Slovak Olympiad III A, 1
Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.
V Soros Olympiad 1998 - 99 (Russia), 9.3
On the coordinate plane, draw a set of points $M(x;y)$, whose coordinates satisfy the equation $$\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.$$
2024 CMIMC Algebra and Number Theory, 8
Compute the number of non-negative integers $k < 2^{20}$ such that $\binom{5k}{k}$ is odd.
[i]Proposed by David Tang[/i]
2012 European Mathematical Cup, 1
Let $ABC$ be a triangle and $Q$ a point on the internal angle bisector of $\angle BAC $. Circle $\omega_1$ is circumscribed to triangle $BAQ$ and intersects the segment $AC$ in point $P \neq C$. Circle $\omega_2$ is circumscribed to the triangle $CQP$. Radius of the cirlce $\omega_1$ is larger than the radius of $\omega_2$. Circle centered at $Q$ with radius $QA$ intersects the circle $\omega_1$ in points $A$ and $A_1$. Circle centered at $Q$ with radius $QC$ intersects $\omega_1$ in points $C_1$ and $C_2$. Prove $\angle A_1BC_1 = \angle C_2PA $.
[i]Proposed by Matija Bucić.[/i]
2015 Balkan MO Shortlist, A5
Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and
$$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$
(Turkey)
2019 Taiwan TST Round 1, 6
Given a triangle $ \triangle ABC $. Denote its incenter and orthocenter by $ I, H $, respectively. If there is a point $ K $ with $$ AH+AK = BH+BK = CH+CK $$ Show that $ H, I, K $ are collinear.
[i]Proposed by Evan Chen[/i]
2011 Purple Comet Problems, 3
Find the sum of all two-digit integers which are both prime and are 1 more than a multiple of 10.
1983 Federal Competition For Advanced Students, P2, 6
Planes $ \pi _1$ and $ \pi _2$ in Euclidean space $ \mathbb{R} ^3$ partition $ S\equal{}\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $ \mathbb{R} ^3$, at least one of the components of $ S$ meets at least three faces of the cube.
2020 Princeton University Math Competition, 15
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
2021 Durer Math Competition Finals, 13
The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?
2021 New Zealand MO, 1
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$. We are also given that $\angle ABC = \angle CDA = 90^o$. Determine the length of the diagonal $BD$.
2017 Junior Balkan Team Selection Tests - Romania, 4
Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that
$$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$
When does the equality hold?
2012 AMC 10, 14
Two equilateral triangles are contained in a square whose side length is $2\sqrt3$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
$ \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \sqrt3\qquad\textbf{(C)}\ 2\sqrt2-1\qquad\textbf{(D)}\ 8\sqrt3-12\qquad\textbf{(E)}\ \frac{4\sqrt3}{3}$
1954 Poland - Second Round, 2
Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $.