[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words : Anya listed the most, Vasya the least . They were awarded points as follows. Each word listed by only one of them scored $2$ points for this child. Each word listed by two of them scored $1$ point for each of these two children. Words listed by all three of them scored $0$ points. Is it possible that Vasya got the highest score, and Anya the lowest? (A Shapovalov)

Three natural numbers $ a,b,c $ with $ \gcd (a,b) =1 $ define in the Diophantine plane a line $ d: ax+by-c=0. $ Prove that: [b]a)[/b] the distance between any two points from $ d $ is at least $ \sqrt{a^2+b^2} . $ [b]b)[/b] the restriction of $ d $ to the first quadrant of the Diophantine plane is a finite line having at most $ 1+\frac{c}{ab} $ elements.

Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b, L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides. (N. Beluhov)

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

(1) 2009 Kansai University entrance exam Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$. (2) 2009 Rikkyo University entrance exam/Science Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$.

Find all triplets of positive integers $a$, $b$ and $c$ such that $a \geq b \geq c$ and $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2$

Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$, $3 \nmid q$, $p^3 = r^3 - q^2$.

Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$

Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Jeff Lin[/i]

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

Let $f$ be a function that satisfies : \[ \displaystyle f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x. \] Find $f(2004)$.

Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*} \max\{1, x_1\} &= x_2 \\ \max\{2, x_2\} &= 2x_3 \\ &\cdots \\ \max\{n, x_n\} &= nx_1. \\ \end{align*}

Connor is thinking of a two-digit number $n$, which satisfies the following properties: [list] [*] If $n>70$, then $n$ is a perfect square. [*] If $n>40$, then $n$ is prime. [*] If $n<80$, then the sum of the digits of $n$ is $14$. [/list] What is Connor's number? [i]Proposed by Connor Gordon[/i]

Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. [asy] size(250); pair A=origin, B=(96,0), C=(96,22), D=(0,22), W=(16,22), X=(20,0), Y=(80,0), Z=(76,22), P=(24,11), Q=(72,11); draw(P--X--A--D--W--P--Q--Y--B--C--Z--Q^^W--Z^^X--Y); dot(A^^B^^C^^D^^P^^Q^^W^^X^^Y^^Z); pair point=(48,11); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$P$", P, dir(point--P)); label("$Q$", Q, dir(point--Q)); label("$W$", W, N); label("$X$", X, S); label("$Y$", Y, S); label("$Z$", Z, N);[/asy]

Answer the following two questions and justify your answers: (a) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$? (b) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+...+2011^{2012}+2012^{2012}$?

For all positive integers $ n$ prove the inequality: $ \left( \frac{1\plus{}(n\plus{}1)^{n\plus{}1}}{n\plus{}2} \right)^{n\minus{}1}>\left( \frac{1\plus{}n^n}{n\plus{}1} \right)^n.$

For a positive integer $n$, let function $f(n)$ denote the number of positive integers $a\leq n$ such that $\gcd(a,n) = \gcd(a+1,n) = 1$. Find the sum of all $n$ such that $f(n)=15$. [i]Proposed by Harry Kim[/i]

Given a natural $n&gt;2$, let $\{ a_1,a_2,...,a_{\phi (n)} \} \subset \mathbb{Z}$ is the Reduced Residue System (RRS) set of modulo $n$ (also known as the set of integers $k$ where $(k,n)=1$ and no pairs are congruent in modulo $n$ ). if write $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{\phi (n)}}=\frac{a}{b}$$ where $a,b \in \mathbb{N}$ and $(a,b)=1$ , then prove that $n|a$. [i](PP-nine)[/i]

The spectators are seated in a row with no empty places. Each is in a seat which does not match the spectator's ticket. An usher can order two spectators in adjacent seats to trade places unless one of them is already seated correctly. Is it true that from any initial arrangement, the spectators can be brought to their correct seats?

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.

Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$. Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if a) $n=2$, $a_1=1, a_2=1000$ b) $n=1000$, $a_i=i, i=1,2\ldots, 1000$ Sorry for my bad English [color=#008000]Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134[/color]

Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient. [i]Cristinel Mortici[/i]

[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$? [b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$? [i]Proposed by Morteza Saghafian[/i]