[u]General Round[/u] [b]p1.[/b] Alice can bake a pie in $5$ minutes. Bob can bake a pie in $6$ minutes. Compute how many more pies Alice can bake than Bob in $60$ minutes. [b]p2.[/b] Ben likes long bike rides. On one ride, he goes biking for six hours. For the first hour, he bikes at a speed of $15$ miles per hour. For the next two hours, he bikes at a speed of $12$ miles per hour. He remembers biking $90$ miles over the six hours. Compute the average speed, in miles per hour, Ben biked during the last three hours of his trip. [b]p3.[/b] Compute the perimeter of a square with area $36$. [b]p4.[/b] Two ants are standing side-by-side. One ant, which is $4$ inches tall, casts a shadow that is $10$ inches long. The other ant is $6$ inches tall. Compute, in inches, the length of the shadow that the taller ant casts. [b]p5.[/b] Compute the number of distinct line segments that can be drawn inside a square such that the endpoints of the segment are on the square and the segment divides the square into two congruent triangles. [b]p6.[/b] Emily has a cylindrical water bottle that can hold $1000\pi$ cubic centimeters of water. Right now, the bottle is holding $100\pi$ cubic centimeters of water, and the height of the water is $1$ centimeter. Compute the radius of the water bottle. [b]p7.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p8.[/b] A sequence an is recursively defined where $a_n = 3(a_{n-1}-1000)$ for $n > 0$. Compute the smallest integer $x$ such that when $a_0 = x$, $a_n > a_0$ for all $n > 0$. [b]p9.[/b] Compute the probability that two random integers, independently chosen and both taking on an integer value between $1$ and $10$ with equal probability, have a prime product. [b]p10.[/b] If $x$ and $y$ are nonnegative integers, both less than or equal to $2$, then we say that $(x, y)$ is a friendly point. Compute the number of unordered triples of friendly points that form triangles with positive area. [b]p11.[/b] Cindy is thinking of a number which is $4$ less than the square of a positive integer. The number has the property that it has two $2$-digit prime factors. What is the smallest possible value of Cindy's number? [b]p12.[/b] Winona can purchase a pencil and two pens for $250$ cents, or two pencils and three pens for $425$ cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens. [b]p13.[/b] Colin has an app on his phone that generates a random integer betwen $1$ and $10$. He generates $10$ random numbers and computes the sum. Compute the number of distinct possible sums Colin can end up with. [b]p14.[/b] A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle. [b]p15.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p16.[/b] A unit square is subdivided into a grid composed of $9$ squares each with sidelength $\frac13$ . A circle is drawn through the centers of the $4$ squares in the outermost corners of the grid. Compute the area of this circle. [b]p17.[/b] There exists exactly one positive value of $k$ such that the line $y = kx$ intersects the parabola $y = x^2 + x + 4$ at exactly one point. Compute the intersection point. [b]p18.[/b] Given an integer $x$, let $f(x)$ be the sum of the digits of $x$. Compute the number of positive integers less than $1000$ where $f(x) = 2$. [b]p19.[/b] Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$? [b]p20.[/b] Compute all real solutions to $16^x + 4^{x+1} - 96 = 0$. [b]p21.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p22.[/b] $ABCDEFGH$ is an equiangular octagon with side lengths $2$, $4\sqrt2$, $1$, $3\sqrt2$, $2$, $3\sqrt2$, $3$, and $2\sqrt2$,in that order. Compute the area of the octagon. [b]p23.[/b] The cubic $f(x) = x^3 +bx^2 +cx+d$ satisfies $f(1) = 3$, $f(2) = 6$, and $f(4) = 12$. Compute $f(3)$. [b]p24.[/b] Given a unit square, two points are chosen uniformly at random within the square. Compute the probability that the line segment connecting those two points touches both diagonals of the square. [b]p25.[/b] Compute the remainder when: $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [u]General Tiebreaker [/u] [b]Tie 1.[/b] Trapezoid $ABCD$ has $AB$ parallel to $CD$, with $\angle ADC = 90^o$. Given that $AD = 5$, $BC = 13$ and $DC = 18$, compute the area of the trapezoid. [b]Tie 2.[/b] The cubic $f(x) = x^3- 7x - 6$ has three distinct roots, $a$, $b$, and $c$. Compute $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ . [b]Tie 3.[/b] Ben flips a fair coin repeatedly. Given that Ben's first coin flip is heads, compute the probability Ben flips two heads in a row before Ben flips two tails in a row. PS. You should use hide for answers.

Let $(a_n)_{n\ge1}$ and $(b_n)_{n\ge1}$ be positive real sequences such that $$\lim_{n\to\infty}\frac{a_{n+1}-a_n}n=a\in\mathbb R^*_+\enspace\text{and}\enspace\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in\mathbb R^*_+$$ Compute $$\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Let $f: [0,\infty)\to\mathbb R$ be a function such that for any $x>0$ the sequence $\{f(nx)\}_{n\geq 0}$ is increasing. a) If the function is also continuous on $[0,1]$ is it true that $f$ is increasing? b) The same question if the function is continuous on $\mathbb Q \cap [0, \infty)$.

Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which: (i) $x_1+ \dots +x_n=0$, and (ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$). Prove that a) $C_n\geq 2$, and b) $C_n=2$ if and only $2 \mid n$.

[b]p1.[/b] Trung has $2$ bells. One bell rings $6$ times per hour and the other bell rings $10$ times per hour. At the start of the hour both bells ring. After how much time will the bells ring again at the same time? Express your answer in hours. [b]p2.[/b] In a soccer tournament there are $n$ teams participating. Each team plays every other team once. The matches can end in a win for one team or in a draw. If the match ends with a win, the winner gets $3$ points and the loser gets $0$. If the match ends in a draw, each team gets $1$ point. At the end of the tournament the total number of points of all the teams is $21$. Let $p$ be the number of points of the team in the first place. Find $n + p$. [b]p3.[/b] What is the largest $3$ digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$? [b]p4.[/b] Let s(n) be the number of quadruplets $(x, y, z, t)$ of positive integers with the property that $n = x + y + z + t$. Find the smallest $n$ such that $s(n) > 2014$. [b]p5.[/b] Consider a decomposition of a $10 \times 10$ chessboard into p disjoint rectangles such that each rectangle contains an integral number of squares and each rectangle contains an equal number of white squares as black squares. Furthermore, each rectangle has different number of squares inside. What is the maximum of $p$? [b]p6.[/b] If two points are selected at random from a straight line segment of length $\pi$, what is the probability that the distance between them is at least $\pi- 1$? [b]p7.[/b] Find the length $n$ of the longest possible geometric progression $a_1, a_2,..,, a_n$ such that the $a_i$ are distinct positive integers between $100$ and $2014$ inclusive. [b]p8.[/b] Feng is standing in front of a $100$ story building with two identical crystal balls. A crystal ball will break if dropped from a certain floor $m$ of the building or higher, but it will not break if it is dropped from a floor lower than $m$. What is the minimum number of times Feng needs to drop a ball in order to guarantee he determined $m$ by the time all the crystal balls break? [b]p9.[/b] Let $A$ and $B$ be disjoint subsets of $\{1, 2,..., 10\}$ such that the product of the elements of $A$ is equal to the sum of the elements in $B$. Find how many such $A$ and $B$ exist. [b]p10.[/b] During the semester, the students in a math class are divided into groups of four such that every two groups have exactly $2$ students in common and no two students are in all the groups together. Find the maximum number of such groups. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom