If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $

Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$ [i]Proposed by Michael Tang[/i]

The set of all points $(x,y)$ in the plane satisfying $x<y$ and $x^3-y^3>x^2-y^2$ has area $A$. What is the value of $A$? [i]Proposed by Adam Bertelli[/i]

Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?

Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?

Without using a calculator, find out which number is greater: $$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

Let $n>100$ be an integer. The numbers $1,2 \ldots, 4n$ are split into $n$ groups of $4$. Prove that there are at least $\frac{(n-6)^2}{2}$ quadruples $(a, b, c, d)$ such that they are all in different groups, $a<b<c<d$ and $c-b \leq |ad-bc|\leq d-a$.

[b]p1.[/b] The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles? [b]p2.[/b] From point $O$ on the plane there are four rays $OA$, $OB$, $OC$ and $OD$ (not necessarily in that order). It is known that $\angle AOB =40^o$, $\angle BOC = 70^o$, $\angle COD = 80^o$. What values can the angle between rays $OA$ and $OD$ take? (The angle between the rays is from $0^o$ to $180^o$.) [b]p3.[/b] Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles. [b]p4.[/b] Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year? [b]p5.[/b] The difference between two four-digit numbers is $7$. How much can the sums of their digits differ? [b]p6.[/b] The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written on the board. In one move you can increase any of the numbers by $3$ or $5$. What is the minimum number of moves you need to make for all the numbers to become equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

We have a set $M$ of real numbers with $|M|>1$ such that for any $x\in M$ we have either $3x-2\in M$ or $-4x+5\in M$. Show that $M$ is infinite.

[u]Round 1[/u] [b]p1.[/b] Find all pairs $(a,b)$ of positive integers with $a > b$ and $a^2 -b^2 =111$. [b]p2.[/b] Alice drives at a constant rate of $2017$ miles per hour. Find all positive values of $x$ such that she can drive a distance of $x^2$ miles in a time of $x$ minutes. [b]p3.[/b] $ABC$ is a right triangle with right angle at $B$ and altitude $BH$ to hypotenuse $AC$. If $AB = 20$ and $BH = 12$, find the area of triangle $\vartriangle ABC$. [u]Round 2[/u] [b]p4.[/b] Regular polygons $P_1$ and $P_2$ have $n_1$ and $n_2$ sides and interior angles $x_1$ and $x_2$, respectively. If $\frac{n_1}{n_2}= \frac75$ and $\frac{x_1}{x_2}=\frac{15}{14}$ , find the ratio of the sum of the interior angles of $P_1$ to the sum of the interior angles of $P_2$. [b]p5.[/b] Joey starts out with a polynomial $f (x) = x^2 +x +1$. Every turn, he either adds or subtracts $1$ from $f$ . What is the probability that after $2017$ turns, $f$ has a real root? [b]p6.[/b] Find the difference between the greatest and least positive integer values $x$ such that $\sqrt[20]{\lfloor \sqrt[17]{x}\rfloor}=1$. [u]Round 3[/u] [b]p7.[/b] Let $ABCD$ be a square and suppose $P$ and $Q$ are points on sides $AB$ and $CD$ respectively such that $\frac{AP}{PB} = \frac{20}{17}$ and $\frac{CQ}{QD}=\frac{17}{20}$ . Suppose that $PQ = 1$. Find the area of square $ABCD$. [b]p8.[/b] If $$\frac{\sum_{n \ge 0} r^n}{\sum_{n \ge 0} r^{2n}}=\frac{1+r +r^2 +r^3 +...}{1+r^2 +r^4 +r^6 +...}=\frac{20}{17},$$ find $r$ . [b]p9.[/b] Let $\overline{abc}$ denote the $3$ digit number with digits $a,b$ and $c$. If $\overline{abc}_{10}$ is divisible by $9$, what is the probability that $\overline{abc}_{40}$ is divisible by $9$? [u]Round 4[/u] [b]p10.[/b] Find the number of factors of $20^{17}$ that are perfect cubes but not perfect squares. [b]p11.[/b] Find the sum of all positive integers $x \le 100$ such that $x^2$ leaves the same remainder as $x$ does upon division by $100$. [b]p12.[/b] Find all $b$ for which the base-$b$ representation of $217$ contains only ones and zeros. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Find all functions $f:\mathbb{N}_0\to\mathbb{N}$ such that [b]1)[/b] \(f(a)\) divides \(a\) for every \(a\in\mathbb{N}_0\), and [b]2)[/b] for all \(a,b,k\in\mathbb{N}_0\) we have \[ f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr). \]

Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.

Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$ $$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$ Note: $\mathbb {Q}$ is the set of rational numbers.

Determine all triples $(a,b,c)$ of real numbers such that $$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$

If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence \[ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0} \] is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)

[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?. [b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there? [b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$. [b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$. [b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles) [b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not? [b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size? [b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$. [b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$ $ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $ $\textbf{(E)}\text{ must be between -1 and +1} $

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.