[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$? [b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$. [b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$. [b]p4.[/b] Circles of radius $7$, $7$, $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices? [b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square. [b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. [b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$. [b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers. [b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] The coins in Merryland all have different integer values: there is a single $1$ cent coin, a single $2$ cent coin, etc. What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed $2021$ cents? [b]p2.[/b] For every positive integer $k$ let $$a_k = \left(\sqrt{\frac{k + 1}{k}}+\frac{\sqrt{k+1}}{k}-\frac{1}{k}-\sqrt{\frac{1}{k}}\right).$$ Evaluate the product $a_4a_5...a_{99}$. Your answer must be as simple as possible. [b]p3.[/b] Prove that for every positive integer $n$ there is a permutation $a_1, a_2, . . . , a_n$ of $1, 2, . . . , n$ for which $j + a_j$ is a power of $2$ for every $j = 1, 2, . . . , n$. [b]p4.[/b] Each point of the $3$-dimensional space is colored one of five different colors: blue, green, orange, red, or yellow, and all five colors are used at least once. Show that there exists a plane somewhere in space which contains four points, no two of which have the same color. [b]p5.[/b] Suppose $a_1 < b_1 < a_2 < b_2 <... < a_n < b_n$ are real numbers. Let $C_n$ be the union of $n$ intervals as below: $$C_n = [a_1, b_1] \cup [a_2, b_2] \cup ... \cup [a_n, b_n].$$ We say $C_n$ is minimal if there is a subset $W$ of real numbers $R$ for which both of the following hold: (a) Every real number $r$ can be written as $r = c + w$ for some $c$ in $C_n$ and some $w$ in $W$, and (b) If $D$ is a subset of $C_n$ for which every real number $r$ can be written as $r = d + w$ for some $d$ in $D$ and some $w$ in $W$, then $D = C_n$. (i) Prove that every interval $C_1 = [a_1, b_1]$ is minimal. (ii) Prove that for every positive integer $n$, the set $C_n$ is minimal PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that we cannot find $171$ binary sequences (sequences of $0$’s and $1$’s), each of length $12$ such that any two of them differ in at least four positions.

[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown? [b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The circumference of circle $B$ is $10$. A square is inscribed in circle $A$. What is the area of that square? [b]p3.[/b] If $x^2 +y^2 = 1$ and $x, y \in R$, let $q$ be the largest possible value of $x+y$ and $p$ be the smallest possible value of $x + y$. Compute $pq$. [b]p4.[/b] Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches $11$ points before the other person can reach $10$ points, then the person who reached $11$ points wins. If instead the score ends up being tied $10$-to-$10$, then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is $70\%$ and the score is currently at $9$-to-$9$. What is the probability that Yizheng wins the game? [b]p5.[/b] The squares on an $8\times 8$ chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is $\#1$, the top-right square is $\#8$, and the bottom-right square is $\#64$). $1$ grain of wheat is placed on square $\#1$, $2$ grains on square $\#2$, $4$ grains on square $\#3$, and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column? [b]p6.[/b] Let $f$ be any function that has the following property: For all real numbers $x$ other than $0$ and $1$, $$f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2.$$ Compute $f(2)$. [b]p7.[/b] Find all solutions of: $$(x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x.$$ [b]p8.[/b] Let $\vartriangle ABC$ be a triangle where $AB = 25$ and $AC = 29$. $C_1$ is a circle that has $AB$ as a diameter and $C_2$ is a circle that has $BC$ as a diameter. $D$ is a point on $C_1$ so that $BD = 15$ and $CD = 21$. $C_1$ and $C_2$ clearly intersect at $B$; let $E$ be the other point where $C_1$ and $C_2$ intersect. Find all possible values of $ED$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

For any positive integer $n$, let [list] [*]$\tau(n)$ denote the number of positive integer divisors of $n$, [*]$\sigma(n)$ denote the sum of the positive integer divisors of $n$, and [*]$\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$. [/list] Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$, $\sigma(n)$, or $\varphi(n)$, respectively. Prove that if the calculator currently displays $a$, then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses. [i]Proposed by Jaedon Whyte.[/i]

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?

Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

Prove that: $$\left\lfloor10^{n+3}\cdot\sqrt{\overline{\underbrace{11\ldots1}_{2n\text{ times}}}}\right\rfloor=\overline{\underbrace{33\ldots3}_{2n\text{ times}}166}$$ [i]Proposed by Ovidiu Pop[/i]

For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$. Prove that the number $a$ can also be written in this form. [b]Additional problems:[/b] [b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form. [b]b)[/b] In how many ways can the integer $n$ be represented in that way?

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

For every nonempty set $M{}$ of integers denote $S(M)$ the sum of all its elements. Let $A=\{a_1,a_2,\ldots,a_{11}\}$ be a set of positive integers with the properties: 1) $a_1<a_2<\ldots<a_{11};$ 2) for every positive integer $n\leq 1500$ there is a subset $M{}$ of $A{}$ for which $S(M)=n.$ Find the smallest possible value of $a_{10}.$

Let $a>1$ be an integer which is not divisible by four. Prove that there are infinitely many primes $p$ of the form $4k-1$ such that $p | a^d-1$ for some $d<\frac{p-1}{2}$

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

[b]p1A.[/b] Compute $$1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ...$$ $$1 - \frac{1}{2^3} + \frac{1}{3^3} - \frac{1}{4^3} + \frac{1}{5^3} - ...$$ [b]p1B.[/b] Real values $a$ and $b$ satisfy $ab = 1$, and both numbers have decimal expansions which repeat every five digits: $$ a = 0.(a_1)(a_2)(a_3)(a_4)(a_5)(a_1)(a_2)(a_3)(a_4)(a_5)...$$ and $$ b = 1.(b_1)(b_2)(b_3)(b_4)(b_5)(b_1)(b_2)(b_3)(b_4)(b_5)...$$ If $a_5 = 1$, find $b_5$. [b]p2.[/b] $P(x) = x^4 - 3x^3 + 4x^2 - 9x + 5$. $Q(x)$ is a $3$rd-degree polynomial whose graph intersects the graph of $P(x)$ at $x = 1$, $2$, $5$, and $10$. Compute $Q(0)$. [b]p3.[/b] Distinct real values $x_1$, $x_2$, $x_3$, $x_4 $all satisfy $ ||x - 3| - 5| = 1.34953$. Find $x_1 + x_2 + x_3 + x_4$. [b]p4.[/b] Triangle $ABC$ has sides $AB = 8$, $BC = 10$, and $CA = 11$. Let $L$ be the locus of points in the interior of triangle $ABC$ which are within one unit of either $A$, $B$, or $C$. Find the area of $L$. [b]p5.[/b] Triangles $ABC$ and $ADE$ are equilateral, and $AD$ is an altitude of $ABC$. The area of the intersection of these triangles is $3$. Find the area of the larger triangle $ABC$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$. Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

Determine all infinite sets $A$ of positive integers with the following propety: If $a,b \in A$ and $a \ge b$ then $\left\lfloor \frac{a}{b} \right\rfloor \in A$