On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

(a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10?$ (b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers differ by $10.$

Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates. I tried to prove that any circle with center at $(0,0)$ has at most one point with coordinates $(a-\sqrt{2},b-\sqrt{3})$;$a,b \in \mathbb{Z}$. So that when we translate the center to $(\sqrt{2},\sqrt{3})$ we have what we wanted to show.

Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that $$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$ and determine when equality holds.

Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.

Each cell of the board with $2021$ rows and $2022$ columns contains exactly one of the three letters $T$, $M$, and $O$ in a way that satisfies each of the following conditions: [list] [*] In total, each letter appears exactly $2021\times 674$ of times on the board. [*] There are no two squares that share a common side and contain the same letter. [*] Any $2\times 2$ square contains all three letters $T$, $M$, and $O$. [/list] Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$

Positive integer numbers $n$ and $k > 1$ are given. Losyash likes some of the cells of the $n \times n$ checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of $2n + 2$, the upper-left corner of which coincides with the upper-left corner of the board (there are $n$ such rectangles in total). Given $n$ and $k$, determine whether Losyash can color each cell he likes in one of $k$ colors so that in any rectangle of interest to him the number of cells of any two colors differ by no more than $1$.

Let $f(x)$ be a bounded continuous function on $[0,+\infty)$. Prove that every solutions of the equation $y''+14y'+13y=f(x)$ are bounded continuous functions on $[0,+\infty)$

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]

Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: [list] [*]each row contains all ice cream flavors, and [*]each column has different sizes of ice cream cup. [/list]Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.

The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?

$11$ girls and $n$ boys went for mushrooms. They have found $n^2+9n -2$ in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys? (A. Tolpygo, Kiev)

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and $H$ is the orthocenter. The angle bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola. I. Voronovich

4. Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space. If point $Q$ move to points on S expect $(0,0,2)$. Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$. Find the range of moving $R$, then illustrate it.

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$

Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.