Pupils of a school are divided into $112$ groups, of $11$ members each. Any two groups have exactly one common pupil. Prove that: a) there is a pupil that belongs to at least $12$ groups. b) there is a pupil that belongs to all the groups.

Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$. Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$. After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take?

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the inequality \[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]

If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function. [For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]

A set consisting of $n$ points of a plane is called a [i]bosonti $n$-point[/i] if any three of its points are located in vertices of an isosceles triangle. Find all positive integers $n$ for which there exists a bosonti $n$-point.

Points $A,B$ are on circle $\omega$. Points $C$ and $D$ are moved on the arc $AB$, such that $CD$ has constant length. $I_1,I_2$ - incenters of $ABC$ and $ABD$. Prove that line $I_1I_2$ is tangent to some fixed circle.

Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.

For real numbers $x$ and $y$, suppose that $|x| - |y| = 20$ and $|x| + |y| = 23$. Compute the sum of all possible distinct values of $|x - y|$.

A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices are always on the $x$- and $y$-axes respectively. Prove that a point within or on the boundary of the square will in general describe a portion of a conic. For what points of the square does this locus degenerate?

Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.

\(ABCD\) is a inscribed quadrilateral. \(P\) is the intersection point of its diagonals. \(O\) is its circumcenter. \(\Gamma\) is the circumcircle of \(ABO\). \(\Delta\) is the circumcircle of \(CDO\). \(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\). \(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\). Show that \(M,N,P\) are collinear.

Let $f(w) = w^3 - rw^2 + sw - \frac{4\sqrt2}{27}$ denote a polynomial, where $r^2 =\left(\frac{8\sqrt2+10}{7}\right) s$. The roots of $f$ correspond to the sides of a right triangle. Compute the smallest possible area of this triangle

If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then $$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).$$

Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.

In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?

In quadrilateral $ABCD$ holds $AB=6$, $AD=4$, $\angle DAB=\angle ABC = 60^{\circ}$ and $\angle ADC = 90^{\circ}$. Find length of diagonals and area of the quadrilateral

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression. Find those numbers?

For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.