Find all positive integers $l$ for which the equation \[ a^3+b^3+ab=(lab+1)(a+b) \] has a solution over positive integers $a,b$.

Let $m,n$ and $N$ be positive integers and $\mathbb{Z}_{N}=\{0,1,\dots,N-1\}$ a set of residues modulo $N$. Consider a table $m\times n$ such that each one of the $mn$ cells has an element of $\mathbb{Z}_{N}$. A [i]move[/i] is choose an element $g\in \mathbb{Z}_{N}$, a cell in the table and add $+g$ to the elements in the same row/column of the chosen cell(the sum is modulo $N$). Prove that if $N$ is coprime with $m-1,n-1,m+n-1$ then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$.

Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.

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

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

Let $n$ be a positive integer. Let $s: \mathbb N \to \{1, \ldots, n\}$ be a function such that $n$ divides $m-s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \ldots$ be a sequence such that $a_0=0$ and \[a_{k}=a_{k-1}+s(k) \text{ for all }k\ge 1.\] Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$. [i]Proposed by N.V. Tejaswi[/i]

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then $\text{(A)}$ there is no such $p$ $\text{(B)}$ there in only one such $p$ $\text{(C)}$ there is more than one such $p$, but finitely many $\text{(D)}$ there are infinitely many such $p$

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented in infinitely many ways of the form $\pm P(1) \pm P(2) \pm \ldots \pm P(m)$, for some positive integer $m$ and certain choices of $\pm$.

Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m,n)$ for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares. Prove that $a$ divides $2b$. [i]Holden Mui[/i]

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$ $$2x + 4y + 17z = 301.$$ [b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$. [b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$. [b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$ $a, b, c, d, e, f$ real. [b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$

[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]

[b]p1.[/b] The following figure represents a rectangular piece of paper $ABCD$ whose dimensions are $4$ inches by $3$ inches. When the paper is folded along the line segment $EF$, the corners $A$ and $C$ coincide. (a) Find the length of segment $EF$. (b) Extend $AD$ and $EF$ so they meet at $G$. Find the area of the triangle $\vartriangle AEG$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/e8844fd37b3b8163f62fcda1300c8d63221f51.png[/img] [b]p2.[/b] (a) Let $p$ be a prime number. If $a, b, c$, and $d$ are distinct integers such that the equation $(x -a)(x - b)(x - c)(x - d) - p^2 = 0$ has an integer solution $r$, show that $(r - a) + (r - b) + (r - c) + (r - d) = 0$. (b) Show that $r$ must be a double root of the equation $(x - a)(x - b)(x - c)(x - d) - p^2 = 0$. [b]p3.[/b] If $\sin x + \sin y + \sin z = 0$ and $\cos x + \cos y + \cos z = 0$, prove the following statements. (a) $\cos (x - y) = -\frac12$ (b) $\cos (\theta - x) + \cos(\theta - y) + \cos (\theta - z) = 0$, for any angle $\theta$. (c) $\sin^2 x + \sin^2 y + \sin^2 z =\frac32$ [b]p4.[/b] Let $|A|$ denote the number of elements in the set $A$. (a) Construct an infinite collection $\{A_i\}$ of infinite subsets of the set of natural numbers such that $|A_i \cap A_j | = 0$ for $i \ne j$. (b) Construct an infinite collection $\{B_i\}$ of infinite subsets of the set of natural numbers such that $|B_i \cap B_j |$ gives a distinct integer for every pair of $i$ and $j$, $i \ne j$. [b]p5.[/b] Consider the equation $x^4 + y^4 = z^5$. (a) Show that the equation has a solution where $x, y$, and $z$ are positive integers. (b) Show that the equation has infinitely many solutions where $x, y$, and $z$ are positive integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Bokos tribus have $2021$ closed chests, we know that every chest have some amount of rupias and some amount of diamonts. They are going to do a deal with Link, that consits that Link will stay with a amount of chests and Bokos with the rest. Before opening the chests, Link has to say the amount of chest that he will stay with. After this the chests open and Link has to choose the chests with the amount that he previously said. Link doesn´t want to make Bokos angry so he wants to say the smallest number of chest that he will stay with, but guaranteeing that he stay with at least with the half of diamonts, and at least the half of the rupias. What number does Link needs to say?

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.