We consider the set E of all fractions $\frac{1}{n}$, where $n$ is a natural number. A maximal arithmetic progression of length $k$ of the set E is an arithmetic progression of $k$ terms such that all its terms belong to the set E, and it is impossible to extend it to the right or to the left with another element of E. For example, $\frac{1}{20}, \frac{1}{8}, \frac{1}{5}$, is an arithmetic progression in E of length $3$, and it is maximal, since to extend it towards to the right you have to add $\frac{11}{40}$, which does not belong to E, and to extend it to the left you have to add $\frac{-1}{40}$ which does not belong to E either. Prove that for every integer $k> 2$, there exists a maximal arithmetic progression of length $k$ of the set E.

Find all solutions of $a^2 + b^2 = n!$ for positive integers $a$, $b$, $n$ with $a \le b$ and $n < 14$.

Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ? [i]Proposed by F. Petrov, K. Sukhov[/i]

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

Find all prime numbers $ p$ and $ q$ such that $ p$ divides $ q\plus{}6$ and $ q$ divides $ p\plus{}6$.

Integers $x,y,z,t$ satisfy $x^2+y^2=z^2+t^2$and$xy=2zt$ prove that $xyzt=0$ Proposed by $M. Abduvaliev$

[u]Round 1[/u] [b]1.1.[/b] A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$. What is the minimum number of groups needed? [b]1.2.[/b] On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has forgotten which is which, so he labels them in random order. The probability that he labels all three countries correctly can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]1.3.[/b] In a class of $40$ sixth graders, the class average for their final test comes out to be $90$ (out of a $100$). However, a student brings up an issue with problem $5$, and $10$ students receive credit for this question, bringing the class average to a $90.75$. How many points was problem $5$ worth? [u]Round 2[/u] [b]2.1.[/b] Compute $1 - 2 + 3 - 4 + ... - 2022 + 2023$. [b]2.2.[/b] In triangle $ABC$, $\angle ABC = 75^o$. Point $D$ lies on side $AC$ such that $BD = CD$ and $\angle BDC$ is a right angle. Compute the measure of $\angle A$. [b]2.3.[/b] Joe is rolling three four-sided dice each labeled with positive integers from $1$ to $4$. The probability the sum of the numbers on the top faces of the dice is $6$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p + q$. [u]Round 3[/u] [b]3.1.[/b] For positive integers $a, b, c, d$ that satisfy $a + b + c + d = 23$, what is the maximum value of $abcd$? [b]3.2.[/b] A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games? [b]3.3.[/b] For a triangle $\vartriangle ABC$, let $M, N, O$ be the midpoints of $AB$, $BC$, $AC$, respectively. Let $P, Q, R$ be points on $AB$, $BC$, $AC$ such that $AP =\frac13 AB$, $BQ =\frac13 BC$, and $CR =\frac13 AC$. The ratio of the areas of $\vartriangle MNO$ and $\vartriangle P QR$ can be expressed as $\frac{m}{n}$ , where $ m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Round 4[/u] [b]4.1.[/b] $2023$ has the special property that leaves a remainder of $1$ when divided by $2$, $21$ when divided by $22$, and $22$ when divided by $23$. Let $n$ equal the lowest integer greater than $2023$ with the above properties. What is $n$? [b]4.2.[/b] Ants $A, B$ are on points $(0, 0)$ and $(3, 3)$ respectively, and ant A is trying to get to $(3, 3)$ while ant $B$ is trying to get to $(0, 0)$. Every second, ant $A$ will either move up or right one with equal probability, and ant $B$ will move down or left one with equal probability. The probability that the ants will meet each other be $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]4.3.[/b] Find the number of trailing zeros of $100!$ in base $ 49$. PS. You should use hide for answers. Rounds 5-9 have been posted [url=https://artofproblemsolving.com/community/c3h3129723p28347714]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$

Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?

A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers. $a)$ Does there exist a meaningful set which consists of $2019$ numbers? $b)$ Does there exist an infinite meaningful set? Note: The geometric mean of the non-negative numbers $a_1, a_2,\cdots, $ $a_n$ is defined as $\sqrt[n]{a_1a_2\cdots a_n} .$

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit number with all digits equal.

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

[b]p1.[/b] If Bob takes $6$ hours to build $4$ houses, how many hours will he take to build $ 12$ houses? [b]p2.[/b] Compute the value of $\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}$. [b]p3.[/b] Given a line $2x + 5y = 170$, find the sum of its $x$- and $y$-intercepts. [b]p4.[/b] In some future year, BmMT will be held on Saturday, November $19$th. In that year, what day of the week will April Fool’s Day (April $1$st) be? [b]p5.[/b] We distribute $78$ penguins among $10$ people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the $10$ people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take? [b]p6.[/b] A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? [b]p7. [/b]Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct? [b]p8.[/b] The pages of a book are consecutively numbered from $1$ through $480$. How many times does the digit $8$ appear in this numbering? [b]p9.[/b] A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length $2$, what is the area of the flower? [b]p10.[/b] There are two non-consecutive positive integers $a, b$ such that $a^2 - b^2 = 291$. Find $a$ and $b$. [b]p11.[/b] Let $ABC$ be an equilateral triangle. Let $P, Q, R$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose the area of triangle $PQR$ is $1$. Among the $6$ points $A, B, C, P, Q, R$, how many distinct triangles with area $1$ have vertices from that set of $6$ points? [b]p12.[/b] A positive integer is said to be binary-emulating if its base three representation consists of only $0$s and $1$s. Determine the sum of the first $15$ binary-emulating numbers. [b]p13.[/b] Professor $X$ can choose to assign homework problems from a set of problems labeled $ 1$ to $30$, inclusive. No two problems in his assignment can share a common divisor greater than $ 1$. What is the maximum number of problems that Professor $X$ can assign? [b]p14.[/b] Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length $5$ and $6$ respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^o$ . Find the area of trapezoid $ABCD$. [b]p15.[/b] Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.) [b]p16.[/b] Let $ABC$ be a triangle and M be the midpoint of $BC$. If $AB = AM = 5$ and $BC = 12$, what is the area of triangle $ABC$? [b]p17. [/b] A positive integer $n$ is called good if it can be written as $5x+ 8y = n$ for positive integers $x, y$. Given that $42$, $43$, $44$, $45$ and $46$ are good, what is the largest n that is not good? [b]p18.[/b] Below is a $ 7 \times 7$ square with each of its unit squares labeled $1$ to $49$ in order. We call a square contained in the figure [i]good [/i] if the sum of the numbers inside it is odd. For example, the entire square is [i]good [/i] because it has an odd sum of $1225$. Determine the number of [i]good [/i] squares in the figure. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [hide][img]https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png[/hide] [b]p19.[/b] A circle of integer radius $ r$ has a chord $PQ$ of length $8$. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $ r$ such that there exist $6$ euphonic chords for $X$? [b]p20.[/b] On planet [i]Silly-Math[/i], two individuals may play a game where they write the number $324000$ on a whiteboard and take turns dividing the number by prime powers – numbers of the form $p^k$ for some prime $p$ and positive integer $k$. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide $324000$ by in order to ensure a win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. Prove that $1+L_{2^{j}}\equiv 0 \pmod{2^{j+1}}$ for $j \geq 0$.

Show that there are no integers $x, y, z$, and $t$ such that $$\sqrt[3]{x^5 + y^5 + z^5 + t^5} = 2016.$$

Let $ x$ be an integer and $ y,z,w$ be odd positive integers. Prove that $ 17$ divides $ x^{y^{z^w}}\minus{}x^{y^z}$.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

Let's call a pair of positive integers $(k,n)$ [i]interesting[/i] if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$ Find the number of interesting pairs $(k,n)$ with $k \leq 100$ [i]M. Karpuk[/i]

How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1,\dots,n\}$ such that $k^4+(\tau(k))^4$ is prime for all $k=1,\dots,n$. Show that $S(n)$ is always a square.

Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$. [list=1] [*] 20 [*] 25 [*] 15 [*] 50 [/list]