Determine all positive real solutions $(a,b)$ to the following system of equations. \begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}

[b]p1.[/b] Evaluate $(2-0)^2 \cdot 3+ \frac{20}{2+3}$ . [b]p2.[/b] Let $x = 11 \cdot 99$ and $y = 9 \cdot 101$. Find the sumof the digits of $x \cdot y$. [b]p3.[/b] A rectangle is cut into two pieces. The ratio between the areas of the two pieces is$ 3 : 1$ and the positive difference between those areas is $20$. What’s the area of the rectangle? [b]p4.[/b] Edgeworth is scared of elevators. He is currently on floor $50$ of a building, and he wants to go down to floor $1$. Edgeworth can go down at most $4$ floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor $1$? [b]p5.[/b] There are $20$ people at a party. Fifteen of those people are normal and $5$ are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party? [b]p6.[/b] Wam and Sang are chewing gum. Gum comes in packages, each package consisting of $14$ sticks of gum. Wam eats $6$ packs and $9$ individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy? [b]p7.[/b] At Lakeside Health School (LHS), $40\%$ of students are male and $60\%$ of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology? [b]p8.[/b] Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with $10$ mL of iced tea and a $2000$ mL cup of water with $10\%$ raspberry iced tea. If he fills up the cup with $20$ more mL of $10\%$ raspberry iced tea water, what percent of the solution will be iced tea? [b]p9.[/b] Tree $1$ starts at height $220$ m and grows continuously at $3$ m per year. Tree $2$ starts at height $20$ m and grows at $5$ m during the first year, $7$ m per during the second year, $9$ m during the third year, and in general $(3+2n)$ m in the nth year. After which year is Tree $2$ taller than Tree $1$? [b]p10.[/b] Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability $\frac{499}{999}$ , tails with probability $\frac{499}{999}$ , and it lands on its side with probability $\frac{1}{999}$ . For each flip of the coin, Leo agrees to give Chris $4$ dollars if it lands on heads, nothing if it lands on tails, and $2$ dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin $17$ times? [b]p11.[/b] Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of $7$, there are $5$ balls that don’t get divided into any pile. If he divides the balls into piles of $11$, there are $9$ balls that aren’t in any pile. If he divides the balls into piles of $13$, there are $11$ balls that aren’t in any pile. What is the minimumnumber of balls Ephram has? [b]p12.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $C A = 5$. Let $F$ be the midpoint of $AB$. Let $E$ be the point on $AC$ such that $EF \parallel BC$. Let CF and $BE$ intersect at $D$. Find $AD$. [b]p13.[/b] Compute the sum of all even positive integers $n \le 1000$ such that: $$lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)$$. [b]p14.[/b] Find the sum of all palindromes with $6$ digits in binary, including those written with leading zeroes. [b]p15.[/b] What is the side length of the smallest square that can entirely contain $3$ non-overlapping unit circles? [b]p16.[/b] Find the sum of the digits in the base $7$ representation of $6250000$. Express your answer in base $10$. [b]p17.[/b] A number $n$ is called sus if $n^4$ is one more than a multiple of $59$. Compute the largest sus number less than $2023$. [b]p18.[/b] Michael chooses real numbers $a$ and $b$ independently and randomly from $(0, 1)$. Given that $a$ and $b$ differ by at most $\frac14$, what is the probability $a$ and $b$ are both greater than $\frac12$ ? [b]p19.[/b] In quadrilateral $ABCD$, $AB = 7$ and $DA = 5$, $BC =CD$, $\angle BAD = 135^o$ and $\angle BCD = 45^o$. Find the area of $ABCD$. [b]p20.[/b] Find the value of $$\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor$$ [b]p21.[/b] Let $a_n$ be the number of words of length $n$ with letters $\{A,B,C,D\}$ that contain an odd number of $A$s. Evaluate $a_6$. [b]p22.[/b] Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are $69$ people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is? [b]p23.[/b] Find $$\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.$$ [b]p24.[/b] Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$ such that $AB = 1$, $\angle B = 75^o$, and $BC =\sqrt2$. Let lines $\ell_1$ and $\ell_2$ be tangent to $\omega$ at $A$ and $C$ respectively. Let $D$ be the intersection of $\ell_1$ and $\ell_2$. Find $\angle ABD$ (in degrees). [b]p25.[/b] Find the sum of the prime factors of $14^6 +27$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that (i) $P(0) = 1$. (ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$. Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.

Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$. Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.

Determine all real solutions $x,y$ to the system of equations $$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$

Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.

Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)

Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $A, B$ and $C$ are real angles such that $$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$ find $$\cos (A)+\cos (B)+\cos (C)$$

For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.

Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.

Find all real $a,b,c,d,e,f$ that satisfy the system $4a = (b + c + d + e)^4$ $4b = (c + d + e + f)^4$ $4c = (d + e + f + a)^4$ $4d = (e + f + a + b)^4$ $4e = (f + a + b + c)^4$ $4f = (a + b + c + d)^4$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+yf(x))=f(x)+xf(y) \quad \text{for all}\ x,y \in\mathbb{R}\]

[u]Round 5[/u] [b]p13.[/b] Determine the number of different circular bracelets can be made with $7$ beads, all either colored red or black. [b]p14.[/b] The product of $260$ and $n$ is a perfect square. The $2020$th least possible positive integer value of $n$ can be written as$ p^{e_1}_1 \cdot p^{e_2}_2\cdot p^{e_3}_3\cdot p^{e_4}_4$ . Find the sum $p_1 +p_2 +p_3 +p_4 +e_1 +e_2 +_e3 +e_4$. [b]p15.[/b] Let $B$ and $C$ be points along the circumference of circle $\omega$. Let $A$ be the intersection of the tangents at $B$ and $C$ and let $D \ne A$ be on $\overrightarrow{AC}$ such that $AC =CD = 6$. Given $\angle BAC = 60^o$, find the distance from point $D$ to the center of $\omega$. [u]Round 6[/u] [b]p16.[/b] Evaluate $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$. [b]p17.[/b] Let $n(A)$ be the number of elements of set $A$ and $||A||$ be the number of subsets of set $A$. Given that $||A||+2||B|| = 2^{2020}$, find the value of $n(B)$. [b]p18.[/b] $a$ and $b$ are positive integers and $8^a9^b$ has $578$ factors. Find $ab$. [u]Round 7[/u] [b]p19.[/b] Determine the probability that a randomly chosen positive integer is relatively prime to $2019$. [b]p20.[/b] A $3$-by-$3$ grid of squares is to be numbered with the digits $1$ through $9$ such that each number is used once and no two even-numbered squares are adjacent. Determine the number of ways to number the grid. [b]p21.[/b] In $\vartriangle ABC$, point $D$ is on $AC$ so that $\frac{AD}{DC}= \frac{1}{13}$ . Let point $E$ be on $BC$, and let $F$ be the intersection of $AE$ and $BD$. If $\frac{DF}{FB}=\frac{2}{7}$ and the area of $\vartriangle DBC$ is $26$, compute the area of $\vartriangle F AB$. [u]Round 8[/u] [b]p22.[/b] A quarter circle with radius $1$ is located on a line with its horizontal base on the line and to the left of the vertical side. It is then rolled to the right until it reaches its original orientation. Determine the distance traveled by the center of the quarter circle. [b]p23.[/b] In $1734$, mathematician Leonhard Euler proved that $\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+...$. With this in mind, calculate the value of $\frac11-\frac14+\frac19-\frac{1}{16}+...$ (the series obtained by negating every other term of the original series). [b]p24.[/b] Billy the biker is competing in a bike show where he can do a variety of tricks. He knows that one trick is worth $2$ points, $1$ trick is worth $3$ points, and 1 is worth $5$ points, but he doesn’t remember which trick is worth what amount. When it’s Billy’s turn to perform, he does $6$ tricks, randomly choosing which trick to do. Compute the sum of all the possible values of points that Billy could receive in total. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples $(a,b,c)$ of nonzero integers for which \[(1-a)(1-b)(1-c)=(1+a)(1+b)(1+c).\]

On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions: (A) Gives a candy to the student sitting on his left or to the student sitting on his right. (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right. At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions. (Two distributions are different if there is a student who has a different number of candy in each of these distributions.) （Forgive my poor English）

Let $a$ be the root of the equation $x^3-x-1=0$. Find an equation of the third degree with integer coefficients whose root is $a^3$.

Let $a, b$, and $c$ be real numbers such that \[\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.\] Prove that \[\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.\]

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

If A,B are invertible and the set {A^k - B^k | k is a natural number} is finite , then there exists a natural number m such that A^m = B^m.

Prove that for any $\lambda > 3$ there is a number $x$ for which $$\sin x + \sin (\lambda x) \ge 1.8.$$

Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation $$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$ for all positive integers $n$.

Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that $1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.

Prove that \[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots +\frac{1}{2019}-\frac{1}{2020}=\frac{1}{1011}+\frac{1}{1012}+\cdots +\frac{1}{2020}\]