The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.

Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

Let $a,b,c$ are non-negative numbers such that $$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$ then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$

Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers a) Given $a_1=a_2$, prove that $3n$ is a perfect square. b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$. (D. Bazylev)

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of triples $(x,a,b)$ where $x$ is a real number and $a,b$ belong to the set $\{1,2,3,4,5,6,7,8,9\}$ such that $$x^2-a\{x\}+b=0.$$ where $\{x\}$ denotes the fractional part of the real number $x$.

Let $ \mathcal{M} $ a set of $ 3n\ge 3 $ planar points such that the maximum distance between two of these points is $ 1 $. Prove that: [b]a)[/b] among any four points,there are two aparted by a distance at most $ \frac{1}{\sqrt{2}} . $ [b]b)[/b] for $ n=2 $ and any $ \epsilon >0, $ it is possible that $ 12 $ or $ 15 $ of the distances between points from $ \mathcal{M} $ lie in the interval $ (1-\epsilon , 1]; $ but any $ 13 $ of the distances can´t be found all in the interval $ \left(\frac{1}{\sqrt 2} ,1\right]. $ [b]c)[/b] there exists a circle of diameter $ \sqrt{6} $ that contains $ \mathcal{M} . $ [b]d)[/b] some two points of $ \mathcal{M} $ are on a distance not exceeding $ \frac{4}{3\sqrt n-\sqrt 3} . $

For each $x$ with $|x|<1$, compute the sum of the series $$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$

Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there?

Given positive integers $ m,n > 1$. Prove that the equation $ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$

$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).

[u]Round 1[/u] [b]p1.[/b] Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”? [b]p2.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p3.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png[/img] [b]p4.[/b] A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of $64$ moves on an $8\times 8$ chessboard so that it visits each square exactly once and returns to its starting position? [img]https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png[/img] [b]p5.[/b] Ann places the integers $1, 2,..., 100$ in a $10 \times 10$ grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least $30$ and no more than $71$. Can he find a strategy to do this, no matter how Ann originally arranged the numbers? [u]Round 2[/u] [b]p6.[/b] Evelyn and Odette are playing a game with a deck of $101$ cards numbered $1$ through $101$. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card $101$ was played against card $2$ on the $10$th move. Prove that this game will never end. [img]https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png[/img] [b]p7.[/b] The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a $10 \times 10$ grid, crushing any teacups on roofs of buildings within a $5 \times 1$ length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear $4$ cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? [img]https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

Find all values of $a$ such that the two polynomials \[x^2+ax-1\qquad\text{and}\qquad x^2-x+a\] share at least 1 root. [i]2018 CCA Math Bonanza Individual Round #7[/i]

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120.000$ Rs. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9.408$ Rs. How many centimeters long is the ruler used by the seller?

Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by $$q(x) = \prod_{j=1}^{2015} p(x + j)$$. We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$ such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.