During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

Let $n$ be a positive integer. We call $C_n$ the number of positive integers $x$ less than $10^n$ such that the sum of the digits of $2x$ is less than the sum of the digits of $x$. Show that $C_n\geq\frac{4}{9}(10^{n}-1)$.

Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$

Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$

Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.

If $f$ is a real-valued function of one real variable which has a continuous derivative on the closed interval $[a,b]$ and for which there is no $x\in [a,b]$ such that $f(x)=f'(x)=0$, then prove that there is a function $g$ with continuous first derivative on $[a,b]$ such that $fg'-f'g$ is positive on $[a,b].$

Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.

Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

Consider a circle with centre $O$, and $3$ points on it, $A,B$ and $C$, such that $\angle {AOB}< \angle {BOC}$. Let $D$ be the midpoint on the arc $AC$ that contains the point $B$. Consider a point $K$ on $BC$ such that $DK \perp BC$. Prove that $AB+BK=KC$.

[b]p1.[/b] The English alphabet consists of $21$ consonants and $5$ vowels. (We count $y$ as a consonant.) (a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be $4$ consecutive consonants. (b) Give a list to show that there need not be $5$ consecutive consonants. (c) Suppose that all the letters are arranged in a circle. Prove that there must be $5$ consecutive consonants. [b]p2.[/b] From the set $\{1,2,3,... , n\}$, $k$ distinct integers are selected at random and arranged in numerical order (lowest to highest). Let $P(i, r, k, n)$ denote the probability that integer $i$ is in position $r$. For example, observe that $P(1, 2, k, n) = 0$. (a) Compute $P(2, 1,6,10)$. (b) Find a general formula for $P(i, r, k, n)$. [b]p3.[/b] (a) Write down a fourth degree polynomial $P(x)$ such that $P(1) = P(-1)$ but $P(2) \ne P(-2)$ (b) Write down a fifth degree polynomial $Q(x)$ such that $Q(1) = Q(-1)$ and $Q(2) = Q(-2)$ but $Q(3) \ne Q(-3)$. (c) Prove that, if a sixth degree polynomial $R(x)$ satisfies $R(1) = R(-1)$, $R(2) = R(-2)$, and $R(3) = R(-3)$, then $R(x) = R(-x)$ for all $x$. [b]p4.[/b] Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number $N$ of distinct values among these sums. (a) Give an example of five numbers yielding the smallest possible value of $N$. What is this value? (b) Give an example of five numbers yielding the largest possible value of $N$. What is this value? (c) Prove that the values of $N$ you obtained in (a) and (b) are the smallest and largest possible ones. [b]p5.[/b] Let $A_1A_2A_3$ be a triangle which is not a right triangle. Prove that there exist circles $C_1$, $C_2$, and $C_3$ such that $C_2$ is tangent to $C_3$ at $A_1$, $C_3$ is tangent to $C_1$ at $A_2$, and $C_1$ is tangent to $C_2$ at $A_3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The set of positive integers is partitioned into finitely many subsets. Show that some subset $S$ has the following property: for every positive integer $n$, $S$ contains infinitely many multiples of $n$.

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$

Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$? [asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0)); draw((0,5)--(10,5)); draw((10,0)--(10,10)); label("$A$",(0,0),SW); label("$B$",(15,0),SE); label("$C$",(15,10),NE); label("$D$",(0,10),NW); dot((0,10)); dot((15,0)); dot((15,10)); dot((0,0)); [/asy] $\textbf{(A) }45\qquad \textbf{(B) }75\qquad \textbf{(C) }100\qquad \textbf{(D) }125\qquad \textbf{(E) }150\qquad$

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

Determine the maximum value of $m$, such that the inequality \[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \] holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$. When does equality occur?

Let $n\geq 2$ and $1 \leq r \leq n$. Consider the set $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$. Compute the sum $\sum_{X \in S_r}X$

Given vectors $\overline a,\overline b,\overline c\in\mathbb R^n$, show that $$(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2$$where $\langle\overline x,\overline y\rangle$ denotes the scalar (inner) product of the vectors $\overline x$ and $\overline y$ and $\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle$.

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$