Found problems: 85335
2014 AMC 10, 9
For real numbers $w$ and $z$,
\[ \frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014. \]
What is $\tfrac{w+z}{w-z}$ ?
${ \textbf{(A)}\ \ -2014\qquad\textbf{(B)}\ \frac{-1}{2014}\qquad\textbf{(C)}\ \frac{1}{2014}\qquad\textbf{(D)}}\ 1\qquad\textbf{(E)}\ 2014$
1994 Nordic, 1
Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that $OA_1 + OB_1 + OC_1 < a$.
1998 IberoAmerican, 1
Given 98 points in a circle. Mary and Joseph play alternatively in the next way:
- Each one draw a segment joining two points that have not been joined before.
The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.
2005 QEDMO 1st, 9 (G3)
Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$.
Darij
2008 Purple Comet Problems, 1
Find the greatest prime factor of the sum of the two largest two-digit prime numbers.
1967 IMO Longlists, 1
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
2018-2019 SDML (High School), 6
Find the largest prime $p$ less than $210$ such that the number $210 - p$ is composite.
2004 AMC 8, 1
On a map, a 12-centimeter length represents $72$ kilometers. How many kilometers does a 17-centimeter length represent?
$\textbf{(A)}\ 6\qquad
\textbf{(B)}\ 102\qquad
\textbf{(C)}\ 204\qquad
\textbf{(D)}\ 864\qquad
\textbf{(E)}\ 1224$
2010 Hanoi Open Mathematics Competitions, 6
Find the greatest integer less than $(2 +\sqrt3)^5$ .
(A): $721$ (B): $722$ (C): $723$ (D): $724$ (E) None of the above.
2014 ASDAN Math Tournament, 3
A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
1995 Poland - First Round, 12
Find out whether there exist two congruent cubes with a common center such that each face of one cube has a common point with each face of the other.
1964 Swedish Mathematical Competition, 5
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.
2000 May Olympiad, 3
Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.
1969 Kurschak Competition, 2
A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.
Russian TST 2019, P2
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2008 Stanford Mathematics Tournament, 16
Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$-rotational symmetry about a point $ P$—that is, if you rotate it $ 120^\circ$ about $ P$, it doesn't change. If $ PX\equal{}1$, find the area of triangle $ \triangle{GHX}$.
2018 CCA Math Bonanza, TB1
What is the maximum number of diagonals of a regular $12$-gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular.
[i]2018 CCA Math Bonanza Tiebreaker Round #1[/i]
1987 AMC 8, 22
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between
[asy]
pair A,B,C,D;
A=(0,4); B=(3,4); C=(3,0); D=origin;
draw(circle(D,5));
fill((0,5)..(1.5,4.7697)..B--A--cycle,black);
fill(B..(4,3)..(5,0)--C--cycle,black);
draw((0,5)--D--(5,0));
label("A",A,NW);
label("B",B,NE);
label("C",C,S);
label("D",D,SW);
[/asy]
$\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$
2006 Victor Vâlcovici, 3
Let be a natural number $ n $ and a matrix $ A\in\mathcal{M}_n(\mathbb{R}) $ having the property that sum of the squares of all its elements is strictly less than $ 1. $ Prove that the matrices $ I\pm A $ are invertible.
2018 Korea USCM, 1
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.
Kvant 2022, M2703
Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.
2002 JBMO ShortLists, 3
Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality:
$ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$
Jury's variant:
Prove the same, but with $ abc\equal{}2$
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2001 German National Olympiad, 2
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.
VI Soros Olympiad 1999 - 2000 (Russia), 10.5
For what values of $k\ge2$ can the set of natural numbers be colored in $k$ colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?