How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities? $$a^2 + b^2 < 16$$ $$a^2 + b^2 < 8a$$ $$a^2 + b^2 < 8b$$

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions: a) $f(0)=0$; b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$; c) $f(1-x)=1-f(x)$. Determine $f\left(\frac{18}{1991}\right)$.

The set of real numbers $ x$ for which \[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\] is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals? $ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Alisha has $6$ cupcakes and Tyrone has $10$ brownies. Tyrone gives some of his brownies to Alisha so that she has three times as many desserts as Tyrone. How many desserts did Tyrone give to Alisha? [b]p2.[/b] Bisky adds one to her favorite number. She then divides the result by $2$, and gets $56$. What is her favorite number? [b]p3.[/b] What is the maximum number of points at which a circle and a square can intersect? [b]p4.[/b] An integer $N$ leaves a remainder of 66 when divided by $120$. Find the remainder when $N$ is divided by $24$. [b]p5.[/b] $7$ people are chosen to run for student council. How many ways are there to pick $1$ president, $1$ vice president, and $1$ secretary? [b]p6.[/b] Anya, Beth, Chloe, and Dmitri are all close friends, and like to make group chats to talk. How many group chats can be made if Dmitri, the gossip, must always be in the group chat and Anya is never included in them? Group chats must have more than one person. [b]p7.[/b] There exists a telephone pole of height $24$ feet. From the top of this pole, there are two wires reaching the ground in opposite directions, with one wire $25$ feet, and the other wire 40 feet. What is the distance (in feet) between the places where the wires hit the ground? [b]p8.[/b] Tarik is dressing up for a job-interview. He can wear a chill, business, or casual outfit. If he wears a chill oufit, he must wear a t-shirt, shorts, and flip-flops. He has eight of the first, seven of the second, and three of the third. If he wears a business outfit, he must wear a blazer, a tie, and khakis; he has two of the first, six of the second, and five of the third; finally, he can also choose the casual style, for which he has three hoodies, nine jeans, and two pairs of sneakers. How many different combinations are there for his interview? [b]p9.[/b] If a non-degenerate triangle has sides $11$ and $13$, what is the sum of all possibilities for the third side length, given that the third side has integral length? [b]p10.[/b] An unknown disease is spreading fast. For every person who has the this illness, it is spread on to $3$ new people each day. If Mary is the only person with this illness at the start of Monday, how many people will have contracted the illness at the end of Thursday? [b]p11.[/b] Gob the giant takes a walk around the equator on Mars, completing one lap around Mars. If Gob’s head is $\frac{13}{\pi}$ meters above his feet, how much farther (in meters) did his head travel than his feet? [b]p12.[/b] $2022$ leaves a remainder of $2$, $6$, $9$, and $7$ when divided by $4$, $7$, $11$, and $13$ respectively. What is the next positive integer which has the same remainders to these divisors? [b]p13.[/b] In triangle $ABC$, $AB = 20$, $BC = 21$, and $AC = 29$. Let D be a point on $AC$ such that $\angle ABD = 45^o$. If the length of $AD$ can be represented as $\frac{a}{b}$ , what is $a + b$? [b]p14.[/b] Find the number of primes less than $100$ such that when $1$ is added to the prime, the resulting number has $3$ divisors. [b]p15.[/b] What is the coefficient of the term $a^4z^3$ in the expanded form of $(z - 2a)^7$? [b]p16.[/b] Let $\ell$ and $m$ be lines with slopes $-2$, $1$ respectively. Compute $|s_1 \cdot s_2|$ if $s_1$, $s_2$ represent the slopes of the two distinct angle bisectors of $\ell$ and $m$. [b]p17.[/b] R1D2, Lord Byron, and Ryon are creatures from various planets. They are collecting monkeys for King Avanish, who only understands octal (base $8$). R1D2 only understands binary (base $2$), Lord Byron only understands quarternary (base $4$), and Ryon only understands decimal (base $10$). R1D2 says he has $101010101$ monkeys and adds his monkey to the pile. Lord Byron says he has $3231$ monkeys and adds them to the pile. Ryon says he has $576$ monkeys and adds them to the pile. If King Avanish says he has $x$ monkeys, what is the value of $x$? [b]p18.[/b] A quadrilateral is defined by the origin, $(3, 0)$, $(0, 10)$, and the vertex of the graph of $y = x^2 -8x+22$. What is the area of this quadrilateral? [b]p19.[/b] There is a sphere-container, filled to the brim with fruit punch, of diameter $6$. The contents of this container are poured into a rectangular prism container, again filled to the brim, of dimensions $2\pi$ by $4$ by $3$. However, there is an excess amount in the original container. If all the excess drink is poured into conical containers with diameter $4$ and height $3$, how many containers will be used? [b]p20.[/b] Brian is shooting arrows at a target, made of concurrent circles of radius $1$, $2$, $3$, and $4$. He gets $10$ points for hitting the innermost circle, $8$ for hitting between the smallest and second smallest circles, $5$ for between the second and third smallest circles, $2$ points for between the third smallest and outermost circle, and no points for missing the target. Assume for each shot he takes, there is a $20\%$ chance Brian will miss the target, but otherwise the chances of hitting each target are proportional to the area of the region. The chance that after three shots, Brian will have scored $15$ points can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p21.[/b] What is the largest possible integer value of $n$ such that $\frac{2n^3+n^2+7n-15}{2n+1}$ is an integer? [b]p22.[/b] Let $f(x, y) = x^3 + x^2y + xy^2 + y^3$. Compute $f(0, 2) + f(1, 3) +... f(9, 11).$ [b]p23.[/b] Let $\vartriangle ABC$ be a triangle. Let $AM$ be a median from $A$. Let the perpendicular bisector of segment $\overline{AM}$ meet $AB$ and $AC$ at $D$, $E$ respectively. Given that $AE = 7$, $ME = MC$, and $BDEC$ is cyclic, then compute $AM^2$. [b]p24.[/b] Compute the number of ordered triples of positive integers $(a, b, c)$ such that $a \le 10$, $b \le 11$, $c \le 12$ and $a > b - 1$ and $b > c - 1$. [b]p25.[/b] For a positive integer $n$, denote by $\sigma (n)$ the the sum of the positive integer divisors of $n$. Given that $n + \sigma (n)$ is odd, how many possible values of $n$ are there from $1$ to $2022$, inclusive? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.

[b]p1.[/b] Sherry starts with a three-digit positive integer. She subtracts $7$ from it, then multiplies the result by $7$, and then adds $7$ to that. If she ends up with $2023$, what number did she start with? [b]p2.[/b] Square $ABCD$ has side length $1$. Point $X$ lies on $\overline{AB}$ such that $\frac{AX}{XB} = 2$, and point $Y$ lies on $\overline{DX}$ such that $\frac{DY}{YX} = 3$. Compute the area of triangle $DAY$ . [b]p3.[/b] A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$? PS. You should use hide for answers.

Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$). [i]2016 CCA Math Bonanza Individual #9[/i]

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

Do there exist a set $A\subset [0,1]$ such that $(a)$ $A$ is a finite union of segments of total length $\frac{1}{2}$, $(b)$ The symmetric difference of $A$ and $B:=A/2\cup(A/2+1/2)$ is a union of segments of the total length less than $\frac{1}{10000}$?

The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that \[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]

Let $a$, $b$, $c$ be the real numbers. It is true, that $a + b$, $b + c$ and $c + a$ are three consecutive integers, in a certain order, and the smallest of them is odd. Prove that the numbers $a$, $b$, $c$ are also consecutive integers in a certain order.

Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$

Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$. The set $M$ is completed with new fractions according to the rule: take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc. Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.

Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\tfrac 12$.

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros? [b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find the sum of all possible $f(-2017)$. [b]p3. [/b] What is the sum of prime factors of $1000027$? [b]p4.[/b] Let $$\frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m},$$ where $n, m$ are relatively prime. Find $(m - n)$. [b]p5.[/b] Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ [b]p6.[/b] Triangle $\vartriangle ABC$ has $\angle B = 120^o$, $AB = 1$. Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$. [b]p7. [/b]Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$. Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$? [b]p8.[/b] Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$. What is the minimum possible value of $s$? [b]p9. [/b]Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$? [b]p10.[/b] Yikai wants to know Liang’s secret code which is a $6$-digit integer $x$. Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$. For instance, $d(14) = 5$ and $d(3) = 3$. It is given that $$x + d(x) + d(d(x)) + d(d(d(x))) = 999868.$$ Please find $x$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

The expression $\left(1 + \sqrt[6]{26 + 15\sqrt3} -\sqrt[6]{26 - 15\sqrt3}\right)^6= m + n\sqrt{p}$ , where $m, n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Does there exist a nonnegative integer $a$ for which the equation \[\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a\] has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers? [i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]

Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.

Find all $ x\in \mathbb{Z} $ and $ a\in \mathbb{R} $ satisfying \[\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a \]