$11$ girls and $n$ boys went for mushrooms. They have found $n^2+9n -2$ in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys? (A. Tolpygo, Kiev)

Find all positive integers $k$ satisfying: there is only a finite number of positive integers $n$, such that the positive integer solution $x$ of $xn+1\mid n^2+kn+1$ is not unique.

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

We say an integer number $n \ge 1$ is conservative, if the smallest prime divisor of $(n!)^n+1$ is at most $n+2015$. Decide if the number of conservative numbers is infinite or not.

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

The sequence $\{x_n\}$ is given by $$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.

Let $n$ be a natural number with more than $2021$ digits, none of which are $8$ or $9$. Suppose that $n$ has no common factors with $2021$. Prove that it is possible to increase one of the digits of $n$ by at most $2$ so that the resulting number is a multiple of 2021.

Let $a$ and $b$ be relatively prime positive integers such that \[\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots,\] where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each subsequent term. Compute $a+b$.

Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

An integer $n$ is called [i]apocalyptic[/i] if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$. Find the smallest positive apocalyptic number.

Let $S$ be a finite set of integers. We define $d_2(S)$ and $d_3(S)$ as: $\bullet$ $d_2(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^2-y^2 = a$ $\bullet$ $d_3(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^3-y^3 = a$ (a) Let $m$ be an integer and $S = \{m, m+1, \ldots, m+2019\}$. Prove: $$d_2(S) > \frac{13}{7} d_3(S)$$ (b) Let $S_n = \{1, 2, \ldots, n\}$ with $n$ a positive integer. Prove that there exists a $N$ so that for all $n > N$: $$ d_2(S_n) > 4 \cdot d_3(S_n) $$

Determine whether does exists a triangle with area $2004$ with his sides positive integers.

Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors. [i]Proposed by Ankan Bhattacharya [/i]

The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$. (a) Represent $29 - 12\sqrt5$ as a prefect square. (b) Represent $10 - 6\sqrt3$ as a prefect cube. [b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions. $$x+cy = 1$$ $$cx+9y = 3$$ [b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$. (a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point. (b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation. [b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it. Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?” “Well, Miranda, could you tell me something else?” “Sure”, said Miranda, “my age is an odd number”. [b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway. [img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img] [b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? (b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? PS. You should use hide for answers.

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

Define \[ N=\sum\limits_{k=1}^{60}e_k k^{k^k} \] where $e_k \in \{-1, 1\}$ for each $k$. Prove that $N$ cannot be the fifth power of an integer.

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$.

Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

Suppose that $a,b,c$ are real numbers in $(0,\frac{\pi}{2})$ such that $a+b+c=\frac{\pi}{4}$ and $\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}$ , where $x,y,z$ are positive integer numbers. Find $x,y,z$.

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.