We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.

Given the natural number N, consider triples of different positive integers $(a, b, c)$ such that $a + b + c = N$. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by $K(N)$. Prove that: (a) $K(N) >\frac{N}{6}-1$ (b) $K(N) <\frac{2N}{9}$ (L.D. Kurliandchik, Leningrad)

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.

Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.

Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.

In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.

Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.

Consider a sequence of positive integers $a_1,a_2,a_3,...$ such that $a_1>1$ and $$a_{n+1}=\frac{a_n}{p}+p,$$ where $p$ is the greatest prime factor of $a_n$. Prove that for any choice of $a_1$, the sequence $a_1,a_2,a_3,...$ has an infinite terms that are equal between them.

Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$ holds. Find the minimum value of $k$.

Find the remainder when $2^{1990}$ is divided by $1990.$

[u]Round 5[/u] [b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions: (i) Each digit does not exceed $3$. (ii) The number $1$ cannot appear in two consecutive digits. (iii) The number $2$ cannot appear in two consecutive digits. [b]p14.[/b] Find the sum of all distinct prime factors of $103040301$. [b]p15.[/b] Let $ABCA'B'C'$ be a triangular prism with height $3$ where bases $ABC$ and $A'B'C'$ are equilateral triangles with side length $\sqrt6$. Points $P$ and $Q$ lie inside the prism so that $ABCP$ and $A'B'C'Q$ are regular tetrahedra. The volume of the intersection of these two tetrahedra can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find $m + n$. [u]Round 6[/u] [b]p16.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a^2_n -a_{n-1}a_{n+1} = a_n -a_{n-1}$ for all positive integers $n$. Given that $a_0 = 1$ and $a_1 = 4$, compute the smallest positive integer $k$ such that $a_k$ is an integer multiple of $220$. [b]p17.[/b] Vincent the Bug is on an infinitely long number line. Every minute, he jumps either $2$ units to the right with probability $\frac23$ or $3$ units to the right with probability $\frac13$ . The probability that Vincent never lands exactly $15$ units from where he started can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? [b]p18.[/b] Battler and Beatrice are playing the “Octopus Game.” There are $2022$ boxes lined up in a row, and inside one of the boxes is an octopus. Beatrice knows the location of the octopus, but Battler does not. Each turn, Battler guesses one of the boxes, and Beatrice reveals whether or not the octopus is contained in that box at that time. Between turns, the octopus teleports to an adjacent box and secretly communicates to Beatrice where it teleported to. Find the least positive integer $B$ such that Battler has a strategy to guarantee that he chooses the box containing the octopus in at most $B$ guesses. [u]Round 7[/u] [b]p19.[/b] Given that $f(x) = x^2-2$ the number $f(f(f(f(f(f(f(2.5)))))))$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find the greatest positive integer $n$ such that $2^n$ divides $ab+a+b-1$. [b]p20.[/b] In triangle $ABC$, the shortest distance between a point on the $A$-excircle $\omega$ and a point on the $B$-excircle $\Omega$ is $2$. Given that $AB = 5$, the sum of the circumferences of $\omega$ and $\Omega$ can be written in the form $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Note: The $A$-excircle is defined to be the circle outside triangle $ABC$ that is tangent to the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and to the side $ BC$. The $B$-excircle is defined similarly for vertex $B$.) [b]p21.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a_0 = 1$, $a_1 = 1$, and there exists two fixed integer constants $x$ and $y$ for which $a_{n+2}$ is the remainder when $xa_{n+1}+ya_n$ is divided by $15$ for all nonnegative integers $n$. Let $t$ be the least positive integer such that $a_t = 1$ and $a_{t+1} = 1$ if such an integer exists, and let $t = 0$ if such an integer does not exist. Find the maximal value of t over all possible ordered pairs $(x, y)$. [u]Round 8[/u] [b]p22.[/b] A mystic square is a $3$ by $3$ grid of distinct positive integers such that the least common multiples of the numbers in each row and column are the same. Let M be the least possible maximal element in a mystic square and let $N$ be the number of mystic squares with $M$ as their maximal element. Find $M + N$. [b]p23.[/b] In triangle $ABC$, $AB = 27$, $BC = 23$, and $CA = 34$. Let $X$ and $Y$ be points on sides $ AB$ and $AC$, respectively, such that $BX = 16$ and $CY = 7$. Given that $O$ is the circumcenter of $BXY$ , the value of $CO^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p24.[/b] Alan rolls ten standard fair six-sided dice, and multiplies together the ten numbers he obtains. Given that the probability that Alan’s result is a perfect square is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, compute $a$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949416p26408251]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?

If $ a$ and $ b$ are natural numbers such that $ a\plus{}13b$ is divisible by $ 11$ and $ a\plus{}11b$ is divisible by $ 13$, then find the least possible value of $ a\plus{}b$.

For each positive integers $u$ and $n$, say that $u$ is a [i]friend[/i] of $n$ if and only if there exists a positive integer $N$ that is a multiple of $n$ and the sum of digits of $N$ (in base 10) is equal to $u$. Determine all positive integers $n$ that only finitely many positive integers are not a friend of $n$.

[b]p1.[/b] Compute $2017 + 7201 + 1720 + 172$. [b]p2. [/b]A number is called [i]downhill [/i]if its digits are distinct and in descending order. (For example, $653$ and $8762$ are downhill numbers, but $97721$ is not.) What is the smallest downhill number greater than 86432? [b]p3.[/b] Each vertex of a unit cube is sliced off by a planar cut passing through the midpoints of the three edges containing that vertex. What is the ratio of the number of edges to the number of faces of the resulting solid? [b]p4.[/b] In a square with side length $5$, the four points that divide each side into five equal segments are marked. Including the vertices, there are $20$ marked points in total on the boundary of the square. A pair of distinct points $A$ and $B$ are chosen randomly among the $20$ points. Compute the probability that $AB = 5$. [b]p5.[/b] A positive two-digit integer is one less than five times the sum of its digits. Find the sum of all possible such integers. [b]p6.[/b] Let $$f(x) = 5^{4^{3^{2^{x}}}}.$$ Determine the greatest possible value of $L$ such that $f(x) > L$ for all real numbers $x$. [b]p7.[/b] If $\overline{AAAA}+\overline{BB} = \overline{ABCD}$ for some distinct base-$10$ digits $A, B, C, D$ that are consecutive in some order, determine the value of $ABCD$. (The notation $\overline{ABCD}$ refers to the four-digit integer with thousands digit $A$, hundreds digit $B$, tens digit $C$, and units digit $D$.) [b]p8.[/b] A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube? [b]p9.[/b] Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$. [b]p10.[/b] Find the largest integer $N > 1$ such that it is impossible to divide an equilateral triangle of side length $ 1$ into $N$ smaller equilateral triangles (of possibly different sizes). [b]p11.[/b] Let $f$ and $g$ be two quadratic polynomials. Suppose that $f$ has zeroes $2$ and $7$, $g$ has zeroes $1$ and $ 8$, and $f - g$ has zeroes $4$ and $5$. What is the product of the zeroes of the polynomial $f + g$? [b]p12.[/b] In square $PQRS$, points $A, B, C, D, E$, and $F$ are chosen on segments $PQ$, $QR$, $PR$, $RS$, $SP$, and $PR$, respectively, such that $ABCDEF$ is a regular hexagon. Find the ratio of the area of $ABCDEF$ to the area of $PQRS$. [b]p13.[/b] For positive integers $m$ and $n$, define $f(m, n)$ to be the number of ways to distribute $m$ identical candies to $n$ distinct children so that the number of candies that any two children receive differ by at most $1$. Find the number of positive integers n satisfying the equation $f(2017, n) = f(7102, n)$. [b]p14.[/b] Suppose that real numbers $x$ and $y$ satisfy the equation $$x^4 + 2x^2y^2 + y^4 - 2x^2 + 32xy - 2y^2 + 49 = 0.$$ Find the maximum possible value of $\frac{y}{x}$. [b]p15.[/b] A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that $$\displaystyle P(x)=R(T(x))$$

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \). a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \). b) Does there exist a non-constant bounded sequence with the above properties? (A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)

Natural number $n>1$ is given. Let $I$ be a set of integers that are relatively prime to $n$. Define the function $f:I=>N$. We call a function $k-periodic$ if for any $a,b$ , $f(a)=f(b)$ whenever $ k|a-b $. We know that $f$ is $n-periodic$. Prove that minimal period of $f$ divides all other periods. Example: if $n=6$ and $f(1)=f(5)$ then minimal period is 1, if $f(1)$ is not equal to $f(5)$ then minimal period is 3.

Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum $$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$ (with $1 \le r \le s \le 2016$) is a composite number, but: a) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$; b) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $GCD(a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$? $GCD(x, y)$ denotes the greatest common divisor of $x$, $y$. Proposed by Matija Bucić

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.