[b]p1.[/b] If $x^2 = 7$, what is $x^4 + x^2 + 1$? [b]p2.[/b] Richard and Alex are competing in a $150$-meter race. If Richard runs at a constant speed of $5$ meters per second and Alex runs at a constant speed of $3$ meters per second, how many more seconds does it take for Alex to finish the race? [b]p3.[/b] David and Emma are playing a game with a chest of $100$ gold coins. They alternate turns, taking one gold coin if the chest has an odd number of gold coins or taking exactly half of the gold coins if the chest has an even number of gold coins. The game ends when there are no more gold coins in the chest. If Emma goes first, how many gold coins does Emma have at the end? [b]p4.[/b] What is the only $3$-digit perfect square whose digits are all different and whose units digit is $5$? [b]p5.[/b] In regular pentagon $ABCDE$, let $F$ be the midpoint of $\overline{AB}$, $G$ be the midpoint of $\overline{CD}$, and $H$ be the midpoint of $\overline{AE}$. What is the measure of $\angle FGH$ in degrees? [b]p6.[/b] Water enters at the left end of a pipe at a rate of $1$ liter per $35$ seconds. Some of the water exits the pipe through a leak in the middle. The rest of the water exits from the right end of the pipe at a rate of $1$ liter per $36$ seconds. How many minutes does it take for the pipe to leak a liter of water? [b]p7.[/b] Carson wants to create a wire frame model of a right rectangular prism with a volume of $2022$ cubic centimeters, where strands of wire form the edges of the prism. He wants to use as much wire as possible. If Carson also wants the length, width, and height in centimeters to be distinct whole numbers, how many centimeters of wire does he need to create the prism? [b]p8.[/b] How many ways are there to fill the unit squares of a $3 \times 5$ grid with the digits $1$, $2$, and $3$ such that every pair of squares that share a side differ by exactly $1$? [b]p9.[/b] In pentagon ABCDE, $AB = 54$, $AE = 45$, $DE = 18$, $\angle A = \angle C = \angle E$, $D$ is on line segment $\overline{BE}$, and line $BD$ bisects angle $\angle ABC$, as shown in the diagram below. What is the perimeter of pentagon $ABCDE$? [img]https://cdn.artofproblemsolving.com/attachments/2/0/7c25837bb10b128a1c7a292f6ce8ce3e64b292.png[/img] [b]p10.[/b] If $x$ and $y$ are nonzero real numbers such that $\frac{7}{x} + \frac{8}{y} = 91$ and $\frac{6}{x} + \frac{10}{y} = 89$, what is the value of $x + y$? [b]p11.[/b] Hilda and Marianne play a game with a shued deck of $10$ cards, numbered from $1$ to $10$. Hilda draws five cards, and Marianne picks up the five remaining cards. Hilda observes that she does not have any pair of consecutive cards - that is, no two cards have numbers that differ by exactly $1$. Additionally, the sum of the numbers on Hilda's cards is $1$ less than the sum of the numbers on Marianne's cards. Marianne has exactly one pair of consecutive cards - what is the sum of this pair? [b]p12.[/b] Regular hexagon $AUSTIN$ has side length $2$. Let $M$ be the midpoint of line segment $\overline{ST}$. What is the area of pentagon $MINUS$? [b]p13.[/b] At a collector's store, plushes are either small or large and cost a positive integer number of dollars. All small plushes cost the same price, and all large plushes cost the same price. Two small plushes cost exactly one dollar less than a large plush. During a shopping trip, Isaac buys some plushes from the store for 59 dollars. What is the smallest number of dollars that the small plush could not possibly cost? [b]p14.[/b] Four fair six-sided dice are rolled. What is the probability that the median of the four outcomes is $5$? [b]p15.[/b] Suppose $x_1, x_2,..., x_{2022}$ is a sequence of real numbers such that: $x_1 + x_2 = 1$ $x_2 + x_3 = 2$ $...$ $x_{2021} + x_{2022} = 2021$ If $x_1 + x_{499} + x_{999} + x_{1501} = 222$, then what is the value of $x_{2022}$? [b]p16.[/b] A cone has radius $3$ and height $4$. An infinite number of spheres are placed in the cone in the following way: sphere $C_0$ is placed inside the cone such that it is tangent to the base of the cone and to the curved surface of the cone at more than one point, and for $i \ge 1$, sphere $C_i$ is placed such that it is externally tangent to sphere $C_{i-1}$ and internally tangent to more than one point of the curved surface of the cone. If $V_i$ is the volume of sphere $C_i$, compute $V_0 + V_1 + V_2 + ... $ . [img]https://cdn.artofproblemsolving.com/attachments/b/4/b43e40bb0a5974dd9d656691c14b4ae268b5b5.png[/img] [b]p17.[/b] Call an ordered pair, $(x, y)$, relatable if $x$ and $y$ are positive integers where $y$ divides $3600$, $x$ divides $y$ and $\frac{y}{x}$ is a prime number. For every relatable ordered pair, Leanne wrote down the positive difference of the two terms of the pair. What is the sum of the numbers she wrote down? [b]p18.[/b] Let $r, s$, and $t$ be the three roots of $P(x) = x^3 - 9x - 9$. Compute the value of $(r^3 + r^2 - 10r - 8)(s^3 + s^2 - 10s - 8)(t^3 + t^2 - 10t - 8)$. [b]p19.[/b] Compute the number of ways to color the digits $0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9$ red, blue, or green such that: (a) every prime integer has at least one digit that is not blue, and (b) every composite integer has at least one digit that is not green. Note that $0$ is not composite. For example, since $12$ is composite, either the digit $1$, the digit $2$, or both must be not green. [b]p20.[/b] Pentagon $ABCDE$ has $AB = DE = 4$ and $BC = CD = 9$ with $\angle ABC = \angle CDE = 90^o$, and there exists a circle tangent to all five sides of the pentagon. What is the length of segment $\overline{AE}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

For any function $f:\mathbb{N}\to\mathbb{N}$ we define $P(n)=f(1)f(2)...f(n)$ . Find all functions $f:\mathbb{N}\to\mathbb{N}$ st for each $a,b$ : $$P(a)+P(b) | a! + b!$$

A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$. Find $100\theta$. [i](B. Dejean, 5 points)[/i]

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds \[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \] where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values ​​of $n$?

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.

How many ways are there to paint the squares of a $6 \times 6$ board black or white such that within each $2\times 2$ square on the board, the number of black squares is odd?

Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$. Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.

Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that $$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?