Find the number of positive integers $n$ less than $10000$ such that there are more $4$’s in the digits of $n + 1$ than in the digits of $n$.

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and $ x_1\plus{}...\plus{}x_n \equiv k$ mod 4 a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$. (b) Prove that $ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$ for all positive integers $ n$.

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$, $$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$

Given a positive integer $h$, show that there are a finite number of triangles with integer sides $a, b, c$ and altitude relative to side $c$ equal to $h$ .

[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$. (a) Show that if the equation has a solution, then $n$ is even. (b) Suppose $n$ is divisible by $4$. Show that the equation has a solution. (c) Show that if the equation has a solution, then $n$ is divisible by $4$. [b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. (b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$. [b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$. (b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$ (c) Determine the exact value of the following infinite sum. $$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$ [b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$. (b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete. [b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure). As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$. [img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims. [b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns. Note that it is not required that he return to his starting point. [img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img] [b]p3.[/b] $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track? [b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$. Suggestion: Reflect!! [b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers. Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove: for all positive integers $m, n$ $\frac 1m + \frac 1{m+1} + \dotsb + \frac 1 {m+n} $ is not an integer.

Find all triples $(p,q,r)$ of primes such that $pq=r+1$ and $2(p^2+q^2)=r^2+1$.

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$? [i]Evan Chen and Ankan Bhattacharya[/i]

Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ a_1\equal{}a_2\equal{}1$ and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

[u]Round 1[/u] [b]p1.[/b] How many ways are there to arrange the letters in the word $NEVERLAND$ such that the $2$ $N$’s are adjacent and the two $E$’s are adjacent? Assume that letters that appear the same are not distinct. [b]p2.[/b] In rectangle $ABCD$, $E$ and $F$ are on $AB$ and $CD$, respectively such that $DE = EF = FB$ and $\angle CDE = 45^o$. Find $AB + AD$ given that $AB$ and $AD$ are relatively prime positive integers. [b]p3.[/b] Maisy Airlines sees $n$ takeoffs per day. Find the minimum value of $n$ such that theremust exist two planes that take off within aminute of each other. [u]Round 2[/u] [b]p4.[/b] Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution? [b]p5.[/b] Find the number of ordered pairs $(a,b)$, where $a$ and $b$ are positive integers, such that $$\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.$$ [b]p6.[/b] $25$ balls are arranged in a $5$ by $5$ square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated $180^o$ and still maintain the same configuration can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, find $m+n$. [u]Round 3[/u] [b]p7.[/b] Maisy the ant is on corner $A$ of a $13\times 13\times 13$ box. She needs to get to the opposite corner called $B$. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let $C$ and $D$ be the possible points where she turns on her path. Find $AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2$. [b]p8.[/b] Maisyton has recently built $5$ intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen. [b]p9.[/b] Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$. [u]Round 4[/u] [b]p10.[/b] Triangle $ABC$ is isosceles, with $AB = BC > AC$. Let the angle bisector of $\angle A$ intersect side $\overline{BC}$ at point $D$, and let the altitude from $A$ intersect side $\overline{BC}$ at point $E$. If $\angle A = \angle C= x^o$, then the measure of $\angle DAE$ can be expressed as $(ax -b)^o$, for some constants $a$ and $b$. Find $ab$. [b]p11[/b]. Maisy randomly chooses $4$ integers $w$, $x$, $y$, and $z$, where $w, x, y, z \in \{1,2,3, ... ,2019,2020\}$. Given that the probability that $w^2 + x^2 + y^2 + z^2$ is not divisible by $4$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p12.[/b] Evaluate $$-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),$$ where there are $100$ square root signs. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the following condition: for any positive integers \( m \) and \( n \) such that \( m > n \) and \( m \) is not divisible by \( n \), if we denote by \( r \) the remainder of the division of \( m \) by \( n \), then the remainder of the division of \( f(m) \) by \( n \) is \( f(r) \). [i]Proposed by Mykyta Kharin[/i]

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]