[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if the roots of the equation $ x^3 + ax^2 + bx + c = 0 $, with real coefficients, are real, then the roots of the equation $ 3x^2 + 2ax + b = 0 $ are also real.

Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ ) Find the general term of {$x_n$}

The absolute value of every number in the sequence $\{a_n\}$ is smaller than 2005, and \[a_{n+6}=a_{n+4}+a_{n+2}-a_n.\] holds for all positive integers n. Prove that $\{a_n\}$ is periodic. Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$. [i]Proposed by Gerhard Woeginger, Austria[/i]

Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3 \\ x^4 - y^4 = 8x - y \end{cases}$

Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.

For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.

[u]Round 1[/u] [b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help? [b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles? [b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat? [u]Round 2[/u] [b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make? [b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares? [b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$? [u]Round 3[/u] [b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side. [b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose? [b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$. [u]Round 4[/u] [b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$. [b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right? [b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

[b]p1.[/b] Find the number of zeroes at the end of $20^{17}$. [b]p2.[/b] Express $\frac{1}{\sqrt{20} +\sqrt{17}}$ in simplest radical form. [b]p3.[/b] John draws a square $ABCD$. On side $AB$ he draws point $P$ so that $\frac{BP}{PA}=\frac{1}{20}$ and on side $BC$ he draws point $Q$ such that $\frac{BQ}{QC}=\frac{1}{17}$ . What is the ratio of the area of $\vartriangle PBQ$ to the area of $ABCD$? [b]p4.[/b] Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints? [b]p5.[/b] Alex is playing a game with an unfair coin which has a $\frac15$ chance of flipping heads and a $\frac45$ chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins? [b]p6.[/b] Positive two-digit number $\overline{ab}$ has $8$ divisors. Find the number of divisors of the four-digit number $\overline{abab}$. [b]p7.[/b] Call a positive integer $n$ diagonal if the number of diagonals of a convex $n$-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to $2017$. [b]p8.[/b] There are $4$ houses on a street, with $2$ on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors. [b]p9.[/b] Compute $$|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.$$ [b]p10.[/b] Given points $A,B$ in the coordinate plane, let $A \oplus B$ be the unique point $C$ such that $\overline{AC}$ is parallel to the $x$-axis and $\overline{BC}$ is parallel to the $y$-axis. Find the point $(x, y)$ such that $((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y)$. [b]p11.[/b] In the following subtraction problem, different letters represent different nonzero digits. $\begin{tabular}{ccccc} & M & A & T & H \\ - & & H & A & M \\ \hline & & L & M & T \\ \end{tabular}$ How many ways can the letters be assigned values to satisfy the subtraction problem? [b]p12.[/b] If $m$ and $n$ are integers such that $17n +20m = 2017$, then what is the minimum possible value of $|m-n|$? [b]p13. [/b]Let $f(x)=x^4-3x^3+2x^2+7x-9$. For some complex numbers $a,b,c,d$, it is true that $f (x) = (x^2+ax+b)(x^2+cx +d)$ for all complex numbers $x$. Find $\frac{a}{b}+ \frac{c}{d}$. [b]p14.[/b] A positive integer is called an imposter if it can be expressed in the form $2^a +2^b$ where $a,b$ are non-negative integers and $a \ne b$. How many almost positive integers less than $2017$ are imposters? [b]p15.[/b] Evaluate the infinite sum $$\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...$$ [b]p16.[/b] Each face of a regular tetrahedron is colored either red, green, or blue, each with probability $\frac13$ . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors? [b]p17.[/b] Let $(k,\sqrt{k})$ be the point on the graph of $y=\sqrt{x}$ that is closest to the point $(2017,0)$. Find $k$. [b]p18.[/b] Alice is going to place $2016$ rooks on a $2016 \times 2016$ chessboard where both the rows and columns are labelled $1$ to $2016$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations. [b]p19.[/b] Let $f (n)$ be a function defined recursively across the natural numbers such that $f (1) = 1$ and $f (n) = n^{f (n-1)}$. Find the sum of all positive divisors less than or equal to $15$ of the number $f (7)-1$. [b]p20.[/b] Find the number of ordered pairs of positive integers $(m,n)$ that satisfy $$gcd \,(m,n)+ lcm \,(m,n) = 2017.$$ [b]p21.[/b] Let $\vartriangle ABC$ be a triangle. Let $M$ be the midpoint of $AB$ and let $P$ be the projection of $A$ onto $BC$. If $AB = 20$, and $BC = MC = 17$, compute $BP$. [b]p22.[/b] For positive integers $n$, define the odd parent function, denoted $op(n)$, to be the greatest positive odd divisor of $n$. For example, $op(4) = 1$, $op(5) = 5$, and $op(6) =3$. Find $\sum^{256}_{i=1}op(i).$ [b]p23.[/b] Suppose $\vartriangle ABC$ has sidelengths $AB = 20$ and $AC = 17$. Let $X$ be a point inside $\vartriangle ABC$ such that $BX \perp CX$ and $AX \perp BC$. If $|BX^4 -CX^4|= 2017$, the compute the length of side $BC$. [b]p24.[/b] How many ways can some squares be colored black in a $6 \times 6$ grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct. [b]p25.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = 2$, $AD = 4$, and $\angle ABC = 120^o$. Let $M$ be the midpoint of $BD$. If $\angle AMC = 90^o$, find the length of segment $CD$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$ [i]Proposed by USJL[/i]

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: $xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$

Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$ a) Assume that $k\in S$. Prove that: $k\ge 2$. b) Prove that: $2\in S$.

Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$

For each real parameter $a$, find the number of real solutions to the system $$\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}$$

[b]p1.[/b] Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of $5 : 3 : 2$. If the person who received neither the least nor greatest amount of money was given $42$ dollars, how many dollars did Will distribute in all? [b]p2.[/b] Fan, Zhu, and Ming are driving around a circular track. Fan drives $24$ times as fast as Ming and Zhu drives $9$ times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together? [b]p3.[/b] Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from $1$ to $120$, inclusive. Let the sum of their numbers be $n$. Let the remainder of the operation $\frac{n^2}{4}$ be $r$. If $r$ is $0$ or $1$, Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be $p$. What is $120p$? [b]p4.[/b] Let S be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. How many subsets of $S$ are there such that if $a$ is the number of even numbers in the subset and $b$ is the number of odd numbers in the subset, then $a$ and $b$ are either both odd or both even? By definition, subsets of $S$ are unordered and only contain distinct elements that belong to $S$. [b]p5.[/b] Phillips Academy has five clusters, $WQN$, $WQS$, $PKN$, $FLG$ and $ABB$. The Blue Key heads are going to visit all five clusters in some order, except $WQS$ must be visited before $WQN$. How many total ways can they visit the five clusters? [b]p6.[/b] An astronaut is in a spaceship which is a cube of side length $6$. He can go outside but has to be within a distance of $3$ from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as $m+n\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]p7.[/b] Let $ABCD$ be a square and $E$ be a point in its interior such that $CDE$ is an equilateral triangle. The circumcircle of $CDE$ intersects sides $AD$ and $BC$ at $D$, $F$ and $C$, $G$, respectively. If $AB = 30$, the area of $AFGB$ can be expressed as $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and c is not divisible by the square of any prime. Find $a + b + c$. [b]p8.[/b] Suppose that $x, y, z$ satisfy the equations $$x + y + z = 3$$ $$x^2 + y^2 + z^2 = 3$$ $$x^3 + y^3 + z^3 = 3$$ Let the sum of all possible values of $x$ be $N$. What is $12000N$? [b]p9.[/b] In circle $O$ inscribe triangle $\vartriangle ABC$ so that $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the midpoint of arc $BC$, and let $AD$ intersect $BC$ at $E$. Determine the value of $DE \cdot DA$. [b]p10.[/b] How many ways are there to color the vertices of a regular octagon in $3$ colors such that no two adjacent vertices have the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive real numbers $a_1>a_2>\ldots>a_n$ with sum $s$ are such that the equation $nx^2-sx+1=0$ has a positive root $a_{n+1}$ smaller than $a_n$. Prove that there exists a positive integer $r \leq n$ such that the inequality $a_ra_{r+1} \geq \frac{1}{r}$ holds. [i]M. Zorka[/i]

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.

Integers $a, b$ satisfy the following property: the line $y = 2x + ab$ passes through all intersection points of the two parabolas given by \[y = x^2 + 2x + a, \quad y = 2x^2 +bx,\] which intersect at least once. How many such $(a, b)$ satisfy $|ab| \leq 100$? [i]Proposed by Justin Hsieh[/i]

Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$

Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$. b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$. (A Andjans, Riga) PS. (a) for juniors (b) for seniors