[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops? [b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$ [b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches? [b]p5.[/b] Compute the median of the positive divisors of $9999$. [b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day? [b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$. [b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img] [b]p9.[/b] What is the remainder when $2022$ is divided by $9$? [b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime? [b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag? [b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$ [b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square. [b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task? [b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$. [b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with? [b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img] [b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$. [b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a non-constant polynomial $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}$, we say that $P$ is symmetric if $a_{k}=a_{n-k}$ for every $k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil$. We define the weight of a non-constant polynomial $P \in \mathbb{R}[x]$, denoted by $t(P)$, as the multiplicity of its zero with the highest multiplicity. a) Prove that there exist non-constant, monic, pairwise distinct polynomials $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$, none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric. b) What is the smallest possible value of $t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)$, if $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$ are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?

The quadrilateral $ABCD$ is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral $ABCD$. Prove that $ABCD$ is also a cyclic quadrilateral.

A chord is drawn through the intersection point of the diagonals of an inscribed quadrilateral. It is known that the parts of this chord located outside the quadrilateral have lengths equal to $\frac13$ and $\frac14$ of this chord. In what ratio is this chord divided by the intersection point of the diagonals of the quadrilateral?

Let $N = 2010!+1$. Prove that a) $N$ is not divisible by $4021$; b) $N$ is not divisible by $2027,2029,2039$; c)$ N$ has a prime divisor greater than $2050$.

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]

The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$. $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 18$

How many $ 3$-digit positive integers have digits whose product equals $ 24$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

On the sides $AC$ and $AB$ of an acute $\triangle ABC$ are chosen points $M$ and $N$ respectively. Point $P$ is an intersection point of the segments $BM$ and $CN$ and point $Q$ is an inner point for the quadrilateral $ANPM$, for which $\angle BQC = 90^\circ$ and $\angle BQP = \angle BMQ$. If the quadrilateral $ANPM$ is inscribed in a circle, prove that $\angle QNC = \angle PQC$.

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$. a) Prove that $N$ and $N'$ coincide; b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$; c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

A polynomial $P$ in $n$ variables and real coefficients is called [i]magical[/i] if $P(\mathbb{N}^n)\subset \mathbb{N}$, and moreover the map $P: \mathbb{N}^n \to \mathbb{N}$ is a bijection. Prove that for all positive integers $n$, there are at least \[n!\cdot (C(n)-C(n-1))\] magical polynomials, where $C(n)$ is the $n$-th Catalan number. Here $\mathbb{N}=\{0,1,2,\dots\}$.

Positive numbers $x_1,\ldots,x_n$ satisfy $$\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.$$Prove that $$\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).$$

Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$

Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$. Given that $x_n = 1000$, what is the largest possible value of $n$? [i]Proposed by Evan Chen[/i]

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]

Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends. (a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones. (b) Prove that this is not necessarily true with less than $n^2/4$ types.

Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression $\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$

Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $ Prove that $ A=0. $ [i]Marius Ghergu[/i]

Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$\sum_{i<j}|x_i -x_j |.$$

An integer is digitally divisible if both of the following conditions are fulfilled: $(a)$ None of its digits is zero; $(b)$ It is divisible by the sum of its digits e.g. $322$ is digitally divisible. Show that there are infinitely many digitally divisible integers.