For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. [b](Condition)[/b] For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots? [i]2022 CCA Math Bonanza Lightning Round 2.1[/i]

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.

Let $r > 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue. [list][*]If $r > \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that $AB = \pi$ and $A,B$ have the same color [*]Does the conclusion in (a) hold if $r > \frac{\pi}{2}$?[/list] [i]Proposed by S Muralidharan[/i]

The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is $ \textbf{(A)}\ 112 \qquad \textbf{(B)}\ 100 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 80$

If $a,b\in\mathbb{R},ab\neq 0$, the the possible figure of $ax-y+b=0$ and $bx^2+ay^2=ab$ is [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy80Lzc3NGNjZWNiN2ZjYzIxMTJlYWE5NDlmZmQ0ZjE1NzgwNmNhM2JiLnBuZw==&rn=MTI0MjQ1ODUyMTI0MjQyNTI0MjUyNS5wbmc=[/img][/center]

Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$

Let $ABCD$ be a tetrahedron satisfying i)$\widehat{ACD}+\widehat{BCD}=180^{0}$, and ii)$\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$. Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$.

Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

Let $ABC$ be a triangle with orthocenter $H$. Let the circle $\omega$ have $BC$ as the diameter. Draw tangents $AP$, $AQ$ to the circle $\omega $ at the point $P, Q$ respectively. Prove that $ P,H,Q$ lie on the same line .

For any real number $ x$ prove that: \[ x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]

Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression \[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]

Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.

The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint. (Igor Nagel)

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ? (S Zlobin)

In a tournament, $n$ players participate. Each player plays each other exactly once, with no ties. A player $A$ is said to be [i]champion [/i] if, for every other player $B$, one of the following two situations occurs: (a) $A$ beat $B$; (b) $A$ beat a player $C$ who in turn beat $B$. Prove that in such a tournament there cannot be exactly two champions.

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

[b]p1.[/b] Compute $21 \cdot 21 - 20 \cdot 20$. [b]p2.[/b] A square has side length $2$. If the square is scaled by a factor of $n$, the perimeter of the new square is equal to the area of the original square. Find $10n$. [b]p3.[/b] Kevin has $2$ red marbles and $2$ blue marbles in a box. He randomly grabs two marbles. The probability that they are the same color can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p4.[/b] In a classroom, if the teacher splits the students into groups of $3$ or $4$, there is one student left out. If the students formgroups of $5$, every student is in a group. What is the fewest possible number of students in this classroom? [b]p5.[/b] Find the sum of all positive integer values of $x$ such that $\lfloor \sqrt{x!} \rfloor = x$. [b]p6.[/b] Find the number of positive integer factors of $2021^{(2^0+2^1)} \cdot 1202^{(1^2+0^2)}$. [b]p7.[/b] Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not. [b]p8.[/b] In isosceles $\vartriangle ABC$, $AB = AC$, and $\angle ABC = 72^o$. The bisector of $\angle ABC$ intersects $AC$ at $D$. Given that $BC = 30$, find $AD$. [b]p9.[/b] For an arbitrary positive value of $x$, let $h$ be the area of a regular hexagon with side length $x$ and let $s$ be the area of a square with side length $x$. Find the value of $\left \lfloor \frac{10h}{s} \right \rfloor$. [b]p10.[/b] There is a half-full tub of water with a base of $4$ inches by $5$ inches and a height of $8$ inches. When an infinitely long stick with base $1$ inch by $1$ inch is inserted vertically into the bottom of the tub, the number of inches the water level rises by can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p11.[/b] Find the sum of all $4$-digit numbers with digits that are a permutation of the digits in $2021$. Note that positive integers cannot have first digit $0$. [b]p12.[/b] A $10$-digit base $8$ integer is chosen at random. The probability that it has $30$ digits when written in base $2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p13.[/b] Call a natural number sus if it can be expressed as $k^2 +k +1$ for some positive integer $k$. Find the sum of all sus integers less than $2021$. [b]p14.[/b] In isosceles triangle $ABC$, $D$ is the intersection of $AB$ and the perpendicular to $BC$ through $C$. Given that $CD = 5$ and $AB = BC = 1$, find $\sec^2 \angle ABC$. [b]p15.[/b] Every so often, the minute and hour hands of a clock point in the same direction. The second time this happens after 1:00 is a b minutes later, where a and b are relatively prime positive integers. Find a +b. [b]p16.[/b] The $999$-digit number $N = 123123...123$ is composed of $333$ iterations of the number $123$. Find the least nonnegative integerm such that $N +m$ is a multiple of $101$. [b]p17.[/b] The sum of the reciprocals of the divisors of $2520$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Duncan, Paul, and $6$ Atreides guards are boarding three helicopters. Duncan, Paul, and the guards enter the helicopters at random, with the condition that Duncan and Paul do not enter the same helicopter. Note that not all helicoptersmust be occupied. The probability that Paul has more guards with him in his helicopter than Duncan does can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p19.[/b] Let the minimum possible distance from the origin to the parabola $y = x^2 -2021$ be $d$. The value of d2 can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] In quadrilateral $ABCD$ with interior point $E$ and area $49 \sqrt3$, $\frac{BE}{CE}= 2 \sqrt3$, $\angle ABC = \angle BCD = 90^o$, and $\vartriangle ABC \sim \vartriangle BCD \sim \vartriangle BEC$. The length of $AD$ can be expressed aspn where $n$ is a positive integer. Find $n$. [b]p21.[/b] Find the value of $$\sum^{\infty}_{i=1}\left( \frac{i^2}{2^{i-1}}+\frac{i^2}{2^{i}}+\frac{i^2}{2^{i+1}}\right)=\left( \frac{1^2}{2^{0}}+\frac{1^2}{2^{1}}+\frac{1^2}{2^{2}}\right)+\left( \frac{2^2}{2^{1}}+\frac{2^2}{2^{2}}+\frac{2^2}{2^{3}}\right)+\left( \frac{3^2}{2^{2}}+\frac{2^2}{2^{3}}+\frac{2^2}{2^{4}}\right)+...$$ [b]p22.[/b] Five not necessarily distinct digits are randomly chosen in some order. Let the probability that they form a nondecreasing sequence be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a +b$ is divided by$ 1000$. [b]p23.[/b] Real numbers $a$, $b$, $c$, and d satisfy $$ac -bd = 33$$ $$ad +bc = 56.$$ Given that $a^2 +b^2 = 5$, find the sum of all possible values of $c^2 +d^2$. [b]p24.[/b] Jeff has a fair tetrahedral die with sides labeled $0$, $1$, $2$, and $3$. He continuously rolls the die and record the numbers rolled in that order. For example, if he rolls a $1$, then rolls a $2$, and then rolls a $3$, he writes down $123$. He keeps rolling the die until he writes the substring $2021$. What is the expected number of times he rolls the die? [b]p25.[/b] In triangle $ABC$, $BC = 2\sqrt3$, and $AB = AC = 4\sqrt3$. Circle $\omega$ with center $O$ is tangent to segment $AB$ at $T$ , and $\omega$ is also tangent to ray $CB$ past $B$ at another point. Points $O, T$ , and $C$ are collinear. Let $r$ be the radius of $\omega$. Given that $r^2 = \frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].