Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have \[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \] Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.

Solve the given equation in primes \begin{align*} xyz=1 +2^{y^2+1} \end{align*}

Let $x; y; m; n$ be integers greater than $1$ such that

[b]p1.[/b] There are $2000$ cans of paint. Show that at least one of the following two statements must be true. There are at least $45$ cans of the same color. There are at least $45$ cans all of different colors. [b]p2.[/b] The measures of the $3$ angles of one triangle are all different from each other but are the same as the measures of the $3$ angles of a second triangle. The lengths of $2$ sides of the first triangle are different from each other but are the same as the lengths of $2$ sides of the second triangle. Must the length of the remaining side of the first triangle be the same as the length of the remaining side of the second triangle? If yes, prove it. If not, provide an example. [b]p3.[/b] Consider the sequence $a_1=1$, $a_2=2$, $a_3=5/2$, ... satisfying $a_{n+1}=a_n+(a_n)^{-1}$ for $n>1$. Show that $a_{10000}>141$. [b]p4.[/b] Prove that no matter how $250$ points are placed in a disk of radius $1$, there is a disk of radius $1/10$ that contains at least $3$ of the points. [b]p5.[/b] Prove that: Given any $11$ integers (not necessarily distinct), one can select $6$ of them so that their sum is divisible by $6$. Given any $71$ integers (not necessarily distinct), one can select $36$ of them so that their sum is divisible by $36$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?

Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

There are $168$ primes below $1000$. Then sum of all primes below $1000$ is, $\textbf{(A)}~11555$ $\textbf{(B)}~76127$ $\textbf{(C)}~57298$ $\textbf{(D)}~81722$

Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Today is Saturday, April $25$, $2020$. What is the value of $6 + 4 + 25 + 2020$? [b]p2.[/b] The figure below consists of a $2$ by $3$ grid of squares. How many squares of any size are in the grid? $\begin{tabular}{|l|l|l|} \hline & & \\ \hline & & \\ \hline \end{tabular}$ [b]p3.[/b] James is playing a game. He first rolls a six-sided dice which contains a different number on each side, then randomly picks one of twelve dierent colors, and finally ips a quarter. How many different possible combinations of a number, a color and a flip are there in this game? [b]p4.[/b] What is the sum of the number of diagonals and sides in a regular hexagon? [b]p5.[/b] Mickey Mouse and Minnie Mouse are best friends but they often fight. Each of their fights take up exactly one hour, and they always fight on prime days. For example, they fight on January $2$nd, $3$rd, but not the $4$th. Knowing this, how many total times do Mickey and Minnie fight in the months of April, May and June? [b]p6.[/b] Apple always loved eating watermelons. Normal watermelons have around $13$ black seeds and $25$ brown seeds, whereas strange watermelons had $45$ black seeds and $2$ brown seeds. If Apple bought $14$ normal watermelons and $7$ strange watermelons, then let $a$ be the total number of black seeds and $b$ be the total number of brown seeds. What is $a - b$? [b]p7.[/b] Jerry and Justin both roll a die once. The probability that Jerry's roll is greater than Justin's can be expressed as a fraction in the form $\frac{m}{n}$ in simplified terms. What is $m + n$? [b]p8.[/b] Taylor wants to color the sides of an octagon. What is the minimum number of colors Taylor will need so that no adjacent sides of the octagon will be filled in with the same color? [b]p9.[/b] The point $\frac23$ of the way from ($-6, 8$) to ($-3, 5$) can be expressed as an ordered pair $(a, b)$. What is $|a - b|$? [b]p10.[/b] Mary Price Maddox laughs $7$ times per class. If she teaches $4$ classes a day for the $5$ weekdays every week but doesn't laugh on Wednesdays, then how many times does she laugh after $5$ weeks of teaching? [b]p11.[/b] Let $ABCD$ be a unit square. If $E$ is the midpoint of $AB$ and $F$ lies inside $ABCD$ such that $CFD$ is an equilateral triangle, the positive difference between the area of $CED$ and $CFD$ can be expressed in the form $\frac{a-\sqrt{b}}{c}$ , where $a$, $b$, $c$ are in lowest simplified terms. What is $a + b + c$? [b]p12.[/b] Eddie has musician's syndrome. Whenever a song is a $C$, $A$, or $F$ minor, he begins to cry and his body becomes very stiff. On the other hand, if the song is in $G$ minor, $A$ at major, or $E$ at major, his eyes open wide and he feels like the happiest human being ever alive. There are a total of $24$ keys. How many different possibilities are there in which he cries while playing one song with two distinct keys? [b]p13.[/b] What positive integer must be added to both the numerator and denominator of $\frac{12}{40}$ to make a fraction that is equivalent to $\frac{4}{11}$ ? [b]p14.[/b] The number $0$ is written on the board. Each minute, Gene the genie either multiplies the number on the board by $3$ or $9$, each with equal probability, and then adds either $1$,$2$, or $3$, each with equal probability. Find the expected value of the number after $3$ minutes. [b]p15.[/b] $x$ satisfies $\dfrac{1}{x+ \dfrac{1}{1+\frac{1}{2}}}=\dfrac{1}{2+ \dfrac{1}{1- \dfrac{1}{2+\frac{1}{2}}}}$ Find $x$. [b]p16.[/b] How many different points in a coordinate plane can a bug end up on if the bug starts at the origin and moves one unit to the right, left, up or down every minute for $8$ minutes? [b]p17.[/b] The triplets Addie, Allie, and Annie, are racing against the triplets Bobby, Billy, and Bonnie in a relay race on a track that is $100$ feet long. The first person of each team must run around the entire track twice and tag the second person for the second person to start running. Then, the second person must run once around the entire track and tag the third person, and finally, the third person would only have to run around half the track. Addie and Bob run first, Allie and Billy second, Annie and Bonnie third. Addie, Allie, and Annie run at $50$ feet per minute (ft/m), $25$ ft/m, and $20$ ft/m, respectively. If Bob, Billy, and Bonnie run half as fast as Addie, Allie, and Annie, respectively, then how many minutes will it take Bob, Billy, and Bonnie to finish the race. Assume that everyone runs at a constant rate. [b]p18.[/b] James likes to play with Jane and Jason. If the probability that Jason and Jane play together is $\frac13$, while the probability that James and Jason is $\frac14$ and the probability that James and Jane play together is $\frac15$, then the probability that they all play together is $\frac{\sqrt{p}}{q}$ for positive integers $p$, $q$ where $p$ is not divisible by the square of any prime. Find $p + q$. [b]p19.[/b] Call an integer a near-prime if it is one more than a prime number. Find the sum of all near-primes less than$ 1000$ that are perfect powers. (Note: a perfect power is an integer of the form $n^k$ where $n, k \ge 2$ are integers.) [b]p20.[/b] What is the integer solution to $\sqrt{\frac{2x-6}{x-11}} = \frac{3x-7}{x+6}$ ? [b]p21.[/b] Consider rectangle $ABCD$ with $AB = 12$ and $BC = 4$ with $F$,$G$ trisecting $DC$ so that $F$ is closer to $D$. Then $E$ is on $AB$. We call the intersection of $EF$ and $DB$ $X$, and the intersection of $EG$ and $DB$ is $Y$. If the area of $\vartriangle XY E$ is \frac{8}{15} , then what is the length of $EB$? [b]p22.[/b] The sum $$\sum^{\infty}_{n=2} \frac{1}{4n^2-1}$$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p23.[/b] In square $ABCD$, $M$, $N$, $O$, $P$ are points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, respectively. If $AB = 4$, $AM = BM$ and $DP = 3AP$, the least possible value of $MN + NO + OP$ can be expressed as $\sqrt{x}$ forsome integer x. Find x: [b]p24.[/b] Grand-Ovich the ant is at a vertex of a regular hexagon and he moves to one of the adjacent vertices every minute with equal probability. Let the probability that after $8$ minutes he will have returned to the starting vertex at least once be the common fraction $\frac{a}{b}$ in lowest terms. What is $a + b$? [b]p25.[/b] Find the last two non-zero digits at the end of $2020!$ written as a two digit number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.

Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.

Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled: 1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$ 2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$. Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.

$k$ is an integer. We define the sequence $\{a_n\}_{n=0}^{\infty}$ like this: \[a_0=0,\ \ \ a_1=1,\ \ \ a_n=2ka_{n-1}-(k^2+1)a_{n-2}\ \ (n \geq 2)\] $p$ is a prime number that $p\equiv 3(\mbox{mod}\ 4)$ a) Prove that $a_{n+p^2-1}\equiv a_n(\mbox{mod}\ p)$ b) Prove that $a_{n+p^3-p}\equiv a_n(\mbox{mod}\ p^2)$

Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.

Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

The ternary expansion $x = 0.10101010\cdots$ is given. Give the binary expansion of $x$. Alternatively, transform the binary expansion $y = 0.110110110 \cdots$ into a ternary expansion.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two distinct positive integers are called "relatively consistent" if the larger one can be written as a sum of some distinct positive divisors of the other one. Show that there exist 2018 positive integers such that any two of them are "relatively consistent"

[b]p1.[/b] Evaluate $(2-0)^2 \cdot 3+ \frac{20}{2+3}$ . [b]p2.[/b] Let $x = 11 \cdot 99$ and $y = 9 \cdot 101$. Find the sumof the digits of $x \cdot y$. [b]p3.[/b] A rectangle is cut into two pieces. The ratio between the areas of the two pieces is$ 3 : 1$ and the positive difference between those areas is $20$. What’s the area of the rectangle? [b]p4.[/b] Edgeworth is scared of elevators. He is currently on floor $50$ of a building, and he wants to go down to floor $1$. Edgeworth can go down at most $4$ floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor $1$? [b]p5.[/b] There are $20$ people at a party. Fifteen of those people are normal and $5$ are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party? [b]p6.[/b] Wam and Sang are chewing gum. Gum comes in packages, each package consisting of $14$ sticks of gum. Wam eats $6$ packs and $9$ individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy? [b]p7.[/b] At Lakeside Health School (LHS), $40\%$ of students are male and $60\%$ of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology? [b]p8.[/b] Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with $10$ mL of iced tea and a $2000$ mL cup of water with $10\%$ raspberry iced tea. If he fills up the cup with $20$ more mL of $10\%$ raspberry iced tea water, what percent of the solution will be iced tea? [b]p9.[/b] Tree $1$ starts at height $220$ m and grows continuously at $3$ m per year. Tree $2$ starts at height $20$ m and grows at $5$ m during the first year, $7$ m per during the second year, $9$ m during the third year, and in general $(3+2n)$ m in the nth year. After which year is Tree $2$ taller than Tree $1$? [b]p10.[/b] Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability $\frac{499}{999}$ , tails with probability $\frac{499}{999}$ , and it lands on its side with probability $\frac{1}{999}$ . For each flip of the coin, Leo agrees to give Chris $4$ dollars if it lands on heads, nothing if it lands on tails, and $2$ dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin $17$ times? [b]p11.[/b] Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of $7$, there are $5$ balls that don’t get divided into any pile. If he divides the balls into piles of $11$, there are $9$ balls that aren’t in any pile. If he divides the balls into piles of $13$, there are $11$ balls that aren’t in any pile. What is the minimumnumber of balls Ephram has? [b]p12.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $C A = 5$. Let $F$ be the midpoint of $AB$. Let $E$ be the point on $AC$ such that $EF \parallel BC$. Let CF and $BE$ intersect at $D$. Find $AD$. [b]p13.[/b] Compute the sum of all even positive integers $n \le 1000$ such that: $$lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)$$. [b]p14.[/b] Find the sum of all palindromes with $6$ digits in binary, including those written with leading zeroes. [b]p15.[/b] What is the side length of the smallest square that can entirely contain $3$ non-overlapping unit circles? [b]p16.[/b] Find the sum of the digits in the base $7$ representation of $6250000$. Express your answer in base $10$. [b]p17.[/b] A number $n$ is called sus if $n^4$ is one more than a multiple of $59$. Compute the largest sus number less than $2023$. [b]p18.[/b] Michael chooses real numbers $a$ and $b$ independently and randomly from $(0, 1)$. Given that $a$ and $b$ differ by at most $\frac14$, what is the probability $a$ and $b$ are both greater than $\frac12$ ? [b]p19.[/b] In quadrilateral $ABCD$, $AB = 7$ and $DA = 5$, $BC =CD$, $\angle BAD = 135^o$ and $\angle BCD = 45^o$. Find the area of $ABCD$. [b]p20.[/b] Find the value of $$\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor$$ [b]p21.[/b] Let $a_n$ be the number of words of length $n$ with letters $\{A,B,C,D\}$ that contain an odd number of $A$s. Evaluate $a_6$. [b]p22.[/b] Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are $69$ people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is? [b]p23.[/b] Find $$\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.$$ [b]p24.[/b] Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$ such that $AB = 1$, $\angle B = 75^o$, and $BC =\sqrt2$. Let lines $\ell_1$ and $\ell_2$ be tangent to $\omega$ at $A$ and $C$ respectively. Let $D$ be the intersection of $\ell_1$ and $\ell_2$. Find $\angle ABD$ (in degrees). [b]p25.[/b] Find the sum of the prime factors of $14^6 +27$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].