[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the blackboard, there are numbers $1, 2, \dots , 100$. At each move, Bob erases arbitrary two numbers $a$ and $b$, where $a \ge b > 0$, and writes the single number $\lfloor{a/b}\rfloor$. After $99$ such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that $\lfloor{x}\rfloor$ is the maximum integer not exceeding $x$.)

Suppose that $a_1, a_2, \dots a_{2021}$ are non-negative numbers such that $\sum_{k=1}^{2021} a_k=1$. Prove that $$ \sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3. $$

In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is: $ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $

On an $n\times n$ chart, where $n \geq 4$, stand "$+$" signs in the cells of the main diagonal and "$-$" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations.

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?

What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ? (A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.

Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$

Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Choose points $X,Y,Z$ on the segments $LA,LB,LC,$ respectively such that:$$\angle XBA=\angle YAB,\angle XCA=\angle ZAC.$$ Prove that: $\angle ZBC=\angle YCB.$

Let $ x$ and $ y$ be positive integers such that $ 7x^5 \equal{} 11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and \[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]

Does there exist a set $S$ consisting of rational numbers with the following property: for every integer $n$ there is a unique nonempty, finite subset of $S$, whose elements sum to $n$?

In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled? (S. Markelov)

Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.

$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do [b]not[/b] count as intersections) Find the maximum number one can get.

A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity). (a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$. (b) Show that, for $n$ even, no such tiling is possible

Eva, Per and Anna play with their pocket calculators. They choose different integers and check, whether or not they are divisible by ${11}$. They only look at nine-digit numbers consisting of all the digits ${1, 2, . . . , 9}$. Anna claims that the probability of such a number to be a multiple of ${11}$ is exactly ${1/11}$. Eva has a different opinion: she thinks the probability is less than ${1/11}$. Per thinks the probability is more than ${1/11}$. Who is correct?

Rectangle $ABCD$ has $AB=20$ and $BC=15$. $2$ circles with diameters $AB$ and $AC$ intersect again at point $E$. What is the length of $DE$?

How many subsets, which does not contain two consecutive numbers, are there of the set $\{1,2,\dots ,20\}$ with $8$ elements? $\textbf{(A)}\ {{13}\choose{8}} \qquad\textbf{(B)}\ {{13}\choose{9}} \qquad\textbf{(C)}\ {{14}\choose{8}} \qquad\textbf{(D)}\ {{14}\choose{9}} \qquad\textbf{(E)}\ {{20}\choose{15}}$

Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.

$A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$, $2\sin C\sin (B+C)-\cos B=0$ find the three angles.

Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]