Found problems: 85335
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.
Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$
[i]Proposed by Ada Tsui[/i]
$A$ and $B$ are two opposite vertices of an $n \times n$ board. Within each small square of the board, the diagonal parallel to $AB$ is drawn, so that the board is divided in $2n^{2}$ equal triangles. A coin moves from $A$ to $B$ along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at $B$, there are exactly two seeds in each of the $2n^{2}$ triangles of the board. Determine all the values of $n$ for which such scenario is possible.
The circles $\omega_1$ and $\omega_2$ intersect at $K{}$ and $L{}$. The line $\ell$ touches the circles $\omega_1$ and $\omega_2$ at the points $X{}$ and $Y{}$, respectively. The point $K{}$ lies inside the triangle $XYL$. The line $XK$ intersects $\omega_2$ a second time at the point $Z{}$. Prove that $LY$ is the bisector of the angle $XLZ$.
Let $ABC$ be a triangle with $\angle ABC = 20^{\circ}$ and $\angle ACB = 40^{\circ}$. Let $D$ be a point on $BC$ such that $\angle BAD = \angle DAC$. Let the incircle of triangle $ABC$ touch $BC$ at $E$. Prove that $BD = 2 \cdot CE$.
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
A rectangle \( m \times n \), where \( m \) and \( n \) are natural numbers strictly greater than 1, is partitioned into \( mn \) unit squares, each of which can be colored either black or white. An operation consists of changing the color of all the squares in a row or in a column to the opposite color. Is it possible that, although initially exactly one square is colored black and all the others are white, after a finite number of moves all squares have the same color?
How many parts can space be divided into by :
a) three half-plane?
b) four half-planes?
A nondegenerate triangle with perimeter $1$ has side lengths $a, b,$ and $c$. Prove that \[\left|\frac{a - b}{c + ab}\right| + \left|\frac{b - c}{a + bc}\right| + \left|\frac{c - a}{b + ac}\right| < 2.\]
[i]Proposed by Andrew Wen[/i]
Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.
Find all integer solutions to the equation
\[(x^2-x)(x^2-2x+2)=y^2-1\]
Show that for all positive integers $n$, the number $2^{3^n}+1$ is divisible by $3^{n+1}$.
Let $ABC$ be a triangle for which the shortest side is $AC$. Its inscribed circle with center $I$ touches sides $AB$ and $BC$ in points $D$ and $E$ respectively. Point $M$ is the midpoint of $AC$. Points $F$ and $G$ lie on sides $BC$ and $AB$ respectively so that $FC=CA=AG$. The line through $I$ perpendicular to $MI$ intersects the line segments $AF$ and $CG$ in $P$ and $Q$ respectively. Prove that $AB=BC\Leftrightarrow PD=QE$.
Find the minimum possible length of the sum of $1999$ unit vectors in the coordinate plane whose both coordinates are nonnegative.
There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$
Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and
$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
At the nodes of graph paper, gardeners live; everywhere around them grow flowers. Each flower is to be taken care of by the three gardeners nearest to it. One of the gardeners wishes to know which are the flowers (s)he has to take care of. Sketch the plot of these gardeners.
Proposed by I. F. Sharygin
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that
\[f(x+t) - f(x) = P(x)\]
for all $x \in \mathbb{R}$.
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.
Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$
In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.