Find the rightmost nonzero digit of $ \frac{100!}{5^{20}}$ (here $ n!\equal{}1\cdot2\cdot3\cdot\ldots\cdot n$).

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

Let $\Gamma$ be the circumcircle of an acute triangle $ABC$. The perpendicular line to $AB$ passing through $C$ cuts $AB$ in $D$ and $\Gamma$ again in $E$. The bisector of the angle $C$ cuts $AB$ in $F$ and $\Gamma$ again in $G$. The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I$. Prove that $AI = EB$.

Right $\vartriangle ABC$ has side lengths $6, 8$, and $10$. Find the positive integer $n$ such that the area of the region inside the circumcircle but outside the incircle of $\vartriangle ABC$ is $n\pi$. [img]https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png[/img]

In unequal triangle $ABC$ bisector $AK$ was drawn. Diameter $XY$ of its circumcircle is perpendicular to $AK$ (order of points on circumcircle is $B-X-A-Y-C$). A circle, passing on points $X$ and $Y$, intersect segments $BK$ and $CK$ in points $T$ and $Z$ respectively. Prove that if $KZ=KT$, then $XT \perp YZ$.

In a room are several people, some of which always lie and all others always tell the truth. Their ages are pairwise distinct. Each person says one of the following phrases: "In this room, there is an equal number of truth-sayers older than me and of liars younger than me" or "In this room, there is an equal number of truth-sayers younger than me and of liars older than me" What is the maximum possible number of truth-sayers in the room? Find an example in which this maximum is achieved and prove a higher number is impossible.

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

Find the positive integer $n$ such that $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 + \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$ .

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$. [i]Proposed by Vivian Loh [/i]

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? $ \textbf{(A) }5 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }15 \qquad $

For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$. Evaluate \[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.