Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

Find all positive integers n such that the product $1! \cdot 2! \cdot \cdot \cdot \cdot n!$ is a perfect square

For each positive integer $n$ let $\tau (n)$ be the sum of divisors of $n$. Find all positive integers $k$ for which $\tau (kn - 1) \equiv 0$ (mod $k$) for all positive integers $n$.

Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

How many natural numbers smaller than $ 2017 $ can be uniquely (order of summands are not relevant) written as a sum of three powers of $ 2? $ [i]Andrei Eckstein[/i]

$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite. [b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime. [b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$. [b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$. [b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$. [b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$. [b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power. [b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5[/u] [b]p13.[/b] Vincent the Bug is at the vertex $A$ of square $ABCD$. Each second, he moves to an adjacent vertex with equal probability. The probability that Vincent is again on vertex $A$ after $4$ seconds is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p14.[/b] Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, and $\angle BAC = 60^o$. Let $P$ be a point inside the triangle such that $BP = 1$ and $CP =\sqrt3$, let $x$ equal the area of $APC$. Compute $16x^2$. [b]p15.[/b] Let $n$ be the number of multiples of$ 3$ between $2^{2020}$ and $2^{2021}$. When $n$ is written in base two, how many digits in this representation are $1$? [u]Round 6[/u] [b]p16.[/b] Let $f(n)$ be the least positive integer with exactly n positive integer divisors. Find $\frac{f(200)}{f(50)}$ . [b]p17.[/b] The five points $A, B, C, D$, and $E$ lie in a plane. Vincent the Bug starts at point $A$ and, each minute, chooses a different point uniformly at random and crawls to it. Then the probability that Vincent is back at $A$ after $5$ minutes can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p18.[/b] A circle is divided in the following way. First, four evenly spaced points $A, B, C, D$ are marked on its perimeter. Point $P$ is chosen inside the circle and the circle is cut along the rays $PA$, $PB$, $PC$, $PD$ into four pieces. The piece bounded by $PA$, $PB$, and minor arc $AB$ of the circle has area equal to one fifth of the area of the circle, and the piece bounded by $PB$, $PC$, and minor arc $BC$ has area equal to one third of the area of the circle. Suppose that the ratio between the area of the second largest piece and the area of the circle is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [u]Round 7 [/u] [b]p19.[/b] There exists an integer $n$ such that $|2^n - 5^{50}|$ is minimized. Compute $n$. [b]p20.[/b] For nonnegative integers $a = \overline{a_na_{n-1} ... a_2a_1}$, $b = \overline{b_mb_{m-1} ... b_2b_1}$, define their distance to be $$d(a, b) = \overline{|a_{\max\,\,(m,n)} - b_{\max\,\,(m,n)}||a_{\max\,\,(m,n)-1} - b_{\max\,\,(m,n)-1}|...|a_1 - b_1|}$$ where $a_k = 0$ if $k > n$, $b_k = 0$ if $k > m$. For example, $d(12321, 5067) = 13346$. For how many nonnegative integers $n$ is $d(2021, n) + d(12345, n)$ minimized? [b]p21.[/b] Let $ABCDE$ be a regular pentagon and let $P$ be a point outside the pentagon such that $\angle PEA = 6^o$ and $\angle PDC = 78^o$. Find the degree-measure of $\angle PBD$. [u]Round 8[/u] [b]p22.[/b] What is the least positive integer $n$ such that $\sqrt{n + 3} -\sqrt{n} < 0.02$ ? [b]p23.[/b] What is the greatest prime divisor of $20^4 + 21 \cdot 23 - 6$? [b]p24.[/b] Let $ABCD$ be a parallelogram and let $M$ be the midpoint of $AC$. Suppose the circumcircle of triangle $ABM$ intersects $BC$ again at $E$. Given that $AB = 5\sqrt2$, $AM = 5$, $\angle BAC$ is acute, and the area of $ABCD$ is $70$, what is the length of $DE$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949414p26408213]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$

Let $n\ge3$ be an integer. A regular $n$-gon $P$ is given. We randomly select three distinct vertices of $P$. The probability that these three vertices form an isosceles triangle is $1/m$, where $m$ is an integer. How many such integers $n\le 2024$ are there? \[\rm a. ~674\qquad \mathrm b. ~675\qquad \mathrm c. ~682 \qquad\mathrm d. ~684\qquad\mathrm e. ~685\]

Does there exist a natural number $n$ such that $n!$ ends with exactly $2021$ zeros?

For any positive integer $x$, we set $$ g(x) = \text{ largest odd divisor of } x, $$ $$ f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\ 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases} $$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.

Let $a,b,c$ natural numbers for which $a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2$. Show that $ab, bc, ca$ and $ab + bc + ca$ are perfect squares .

For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities $a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$ $a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$ Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$

Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

Let $P(x)=a_d x^d+\dots+a_1 x+a_0$ be a non-constant polynomial with non-negative integer coefficients having $d$ rational roots.Prove that $$\text{lcm} \left(P(m),P(m+1),\dots,P(n) \right)\geq m \dbinom{n}{m}$$ for all $n>m$ [i](Navid Safaei, Iran)[/i]

For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$

Prove that $\sqrt 3$ is irrational.

[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars? [b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side. [b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap? [b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ . [b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square. [b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$? [b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$? [b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$? [b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$. [b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.