p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$ p3. A function $f$ is defined as in the following table. [img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img] Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$ p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$ p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?

What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge a + d$?

Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?

If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals: $\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$

(a) Find all pairs $(m, n)$ of integers satisfying $m^3-4mn^2=8n^3-2m^2n$ (b) Among such pairs find those for which $m+n^2=3$

Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.

In the $ 6\times4$ grid shown, $ 12$ of the $ 24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $ N$ be the number of shadings with this property. Find the remainder when $ N$ is divided by $ 1000$. [asy]size(100); defaultpen(linewidth(0.7)); int i; for(i=0; i<5; ++i) { draw((i,0)--(i,6)); } for(i=0; i<7; ++i) { draw((0,i)--(4,i)); }[/asy]

Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.

Find all pairs of prime numbers $p, q$ for which there exist positive integers $(m, n)$ such that $$(p+q)^m=(p-q)^n$$.

Let $ABC$ be an acute-angled triangle with circumradius $R$ less than the sides of $ABC$, let $H$ and $O$ be the orthocenter and circuncenter of $ABC$, respectively. The angle bisectors of $\angle ABH$ and $\angle ACH$ intersects in the point $A_1$, analogously define $B_1$ and $C_1$. If $E$ is the midpoint of $HO$, prove that $EA_1+EB_1+EC_1=p-\frac{3R}{2}$ where $p$ is the semiperimeter of $ABC$

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.

$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.

Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$. Prove, that $P$, $Q$, $M$ lies at one line.

Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$?

Two $5\times1$ rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]

The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.

For an $n\times n$ matrix with real entries let $||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}$, where $||\cdot||_{2}$ denotes the Euclidean norm on $\mathbb{R}^{n}$. Assume that an $n\times n$ matrxi $A$ with real entries satisfies $||A^{k}-A^{k-1}||\leq\frac{1}{2002k}$ for all positive integers $k$. Prove that $||A^{k}||\leq 2002$ for all positive integers $k$.

Juku invented an apparatus that can divide any segment into three equal segments. How can you find the midpoint of any segment, using only the Juku made, a ruler and pencil?

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.

Compute the sum of the digits of $1001^{10}$

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] [*]$1\le x,y\le 29$ [*]$29\mid y^2-x^p-26$ [/list]

If $x(y + 1) = 41$ and $x^2(y^2 + 1) = 881$, determine all possible pairs of real numbers $(x,y)$.

[b]p1.[/b] Two proper ordinary fractions are given. The first has a numerator that is $5$ less than the denominator, and the second has a numerator that is $1998$ less than the denominator. Can their sum have a numerator greater than its denominator? [b]p2.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in $365$ days on the next New Year's Eve? [b]p3.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p4.[/b] In the quadrilateral $ABCD$, the extensions of opposite sides $AB$ and $CD$ intersect at an angle of $20^o$; the extensions of opposite sides $BC$ and $AD$ also intersect at an angle of $20^o$. Prove that two angles in this quadrilateral are equal and the other two differ by $40^o$. [b]p5.[/b] Given two positive integers $a$ and $b$. Prove that $a^ab^b\ge a^ab^a.$ [b]p6.[/b] The square is divided by straight lines into $25$ rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $ 17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] In expression $$(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q)$$ opened the brackets. How many members will there be? How many of them will be preceded by a minus sign? [b]p9.[/b] In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of $100$ people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it? [b]p10.[/b] Vasya and Petya play such a game on a $10 \times 10 board$. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right? [img]https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]