Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

On the unit circle $k$ with center $O$, points $A$ and $B$ with $AB=1$ are chosen and unit circles $k_1$ and $k_2$ with centers $A$ and $B$ are drawn. A sequence of circles $(l_n)$ is defined as follows: circle $l_1$ is tangent to $k$ internally at $D_1$ and to $k_1,k_2$ externally, and for $n>1$ circle $l_n$ is tangent to $k_1$ and $k_2$ and to $l_{n-1}$ at $D_n$. For each $n$, compute $d_n=OD_n$ and the radius $r_n$ of $l_n$.

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

Each of the numbers in the set \(A = \{1,2, \cdots, 2017\}\) is colored either red or white. Prove that for \(n \geq 18\), there exists a coloring of the numbers in \(A\) such that any of its n-term arithmetic sequences contains both colors.

A particular coin can land on heads (H), on tails (T), or in the middle (M), each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times.

[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].

The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and $$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$ Find the least positive integer $n$ such that $a_n = 1$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [b]C1.[/b] Mr. Pham flips $2018$ coins. What is the difference between the maximum and minimum number of heads that can appear? [b]C2 / G1.[/b] Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable? [b]C3.[/b] Guang has $10$ cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have? [b]C4.[/b] The ninth edition of Campbell Biology has $1464$ pages. If Chris reads from the beginning of page $426$ to the end of page$449$, what fraction of the book has he read? [b]C5 / G2.[/b] The planet Vriky is a sphere with radius $50$ meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey? [b]C6 / G3.[/b] Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from $0$ to $100$ inclusive. However, according to school policy, if the quarter grade is less than or equal to $50$, then it is bumped up to $50$. What is the probability that Stan’s final quarter grade is $50$? [b]C7 / G5.[/b] What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length $1$ and a square of side length $20$? [b]C8.[/b] You enter the MBMT lottery, where contestants select three different integers from $1$ to $5$ (inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win? [b]C9 / G7.[/b] Find a possible solution $(B, E, T)$ to the equation $THE + MBMT = 2018$, where $T, H, E, M, B$ represent distinct digits from $0$ to $9$. [b]C10.[/b] $ABCD$ is a unit square. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $AD$. $DE$ and $CF$ meet at $G$. Find the area of $\vartriangle EFG$. [b]C11.[/b] The eight numbers $2015$, $2016$, $2017$, $2018$, $2019$, $2020$, $2021$, and $2022$ are split into four groups of two such that the two numbers in each pair differ by a power of $2$. In how many different ways can this be done? [b]C12 / G4.[/b] We define a function f such that for all integers $n, k, x$, we have that $$f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x).$$ If $f(1, k) = 2k$ for all integers $k$, then what is $f(3, 7)$? [b]C13 / G8.[/b] A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be $4$, $79$, $1035$. How long is the longest possible sequence that satisfies these rules? [b]C14 / G11.[/b] $ABC$ is an equilateral triangle of side length $8$. $P$ is a point on side AB. If $AC +CP = 5 \cdot AP$, find $AP$. [b]C15.[/b] What is the value of $(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)$? [b]G6.[/b] An ant is on a coordinate plane. It starts at $(0, 0)$ and takes one step each second in the North, South, East, or West direction. After $5$ steps, what is the probability that the ant is at the point $(2, 1)$? [b]G10.[/b] Find the set of real numbers $S$ so that $$\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).$$ [b]G12.[/b] Given a function $f(x)$ such that $f(a + b) = f(a) + f(b) + 2ab$ and $f(3) = 0$, find $f\left( \frac12 \right)$. [b]G13.[/b] Badville is a city on the infinite Cartesian plane. It has $24$ roads emanating from the origin, with an angle of $15$ degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from $(10, 0)$ to $(3, 3)$. What is the minimum distance he can take, only going on roads? [b]G14.[/b] Team $A$ and Team $B$ are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a $50\%$ chance of scoring $1$ point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team $A$ will score $5$ points before Team $B$ scores any? [b]G15.[/b] The twelve-digit integer $$\overline{A58B3602C91D},$$ where $A, B, C, D$ are digits with $A > 0$, is divisible by $10101$. Find $\overline{ABCD}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. There exist $n$ ordered triples$$(x_1, y_1, z_1), (x_2, y_2, z_2), \dots, (x_n, y_n, z_n)$$where each coordinate is an integer between $1$ and $100$ (inclusive), satisfying the following condition: [center] [i]For every infinite sequence $(a_1, a_2, a_3, \dots)$ of integers between $1$ and $100$, there exist a positive integer $i$ and an index $j$ (with $1 \leqslant j \leqslant n$) such that $(a_i, a_{i+1}, a_{i+2}) = (x_j, y_j, z_j)$.[/i] [/center] Determine the minimum possible value of $n$.

Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that $2a_n-1$ is a perfect square.

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , \[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]

Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$. Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.

The sequence $ (x_n)_{n \geq 1}$ is defined by: $ x_1\equal{}1$ $ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$ Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.