Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

[u]Round 1[/u] [b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists. [b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins? [b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle? [u]Round 2 [/u] [b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing? [b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$? [b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$? [u]Round 3[/u] [b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$? [b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$? [b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer? [u]Round 4[/u] [b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$. [b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers. [b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$ $\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers. [i]Proposed by Navid Safaei, Iran[/i]

Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$ Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.

Let $p{}$ be a prime number. There are $p{}$ integers $a_0,\ldots,a_{p-1}$ around a circle. In one move, it is allowed to select some integer $k{}$ and replace the existing numbers via the operation $a_i\mapsto a_i-a_{i+k}$ where indices are taken modulo $p{}.$ Find all pairs of natural numbers $(m, n)$ with $n>1$ such that for any initial set of $p{}$ numbers, after performing any $m{}$ moves, the resulting $p{}$ numbers will all be divisible by $n{}.$ [i]Proposed by P. Kozhevnikov[/i]

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?

Find all positive integers $n$ for which the equation $$x^3 + y^3 = n! + 4$$ has solutions in integers.

How many triples $(a,b,c)$ of positive integers strictly less than $51$ are there such that $a+b+c$ is divisible by $a, b$, and $c$?

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.

Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

How many reorderings of $2,3,4,5,6$ have the property that every pair of adjacent numbers are relatively prime? [i]2021 CCA Math Bonanza Individual Round #3[/i]

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$ [b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img] [b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$. [b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png [/img] [b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends. [b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that $$a_1 - 2a_2 + a_3 \le 0$$ $$a_2-2a_3 + a_ 4 \le 0$$ $$...$$ $$a_{98}-2a_{99 }+ a_{100} \le 0$$ and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.

Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying: (i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $ (ii) $\ N - a_i$ is a perfect square for $\ i = 1,2,3,4,5,6 $.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccccc} & & & a & b & c & d \\ x & & & & & a & b \\ \hline & & c & d & b & d & b \\ + & c & e & b & f & b & \\ \hline & c & g & a & e & g & b \\ \end{tabular}$ [b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$? [b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$ [b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$