At Durant University, an A grade corresponds to raw scores between $90$ and $100$, and a B grade corresponds to raw scores between $80$ and $90$. Travis has $3$ equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a $90\%$ chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.

Let $x_1$, $x_2$,$x_3$, $y_1$, $y_2$, and $y_3 $ be positive real numbers, such that $x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1$. Prove that $$ x_1y_1 + x_2y_2 + x_3y_3 <1$$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions : 1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$ 2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

Let $a,b$ be positive integers. Prove that the equation $x^2+(a+b)^2x+4ab=1$ has rational solutions if and only if $a=b$. [i]Mihai Opincariu[/i]

p1. $A$ is a set of numbers. The set $A$ is closed to subtraction, meaning that the result of subtracting two numbers in $A$ will be returns a number in $A$ as well. If it is known that two members of $A$ are $4$ and $9$, show that: a. $0\in A$ b. $13 \in A$ c. $74 \in A$ d. Next, list all the members of the set $A$ . p2. $(2, 0, 4, 1)$ is one of the solutions/answers of $x_1+x_2+x_3+x_4=7$. If all solutions belong on the set of not negative integers , specify as many possible solutions/answers from $x_1+x_2+x_3+x_4=7$ p3. Adi is an employee at a textile company on duty save data. One time Adi was asked by the company leadership to prepare data on production increases over five periods. After searched by Adi only found four data on the increase, namely $4\%$, $9\%$, $7\%$, and $5\%$. One more data, namely the $5$th data, was not found. Investigate increase of 5th data production, if Adi only remembers that the arithmetic mean and median of the five data are the same. p4. Find all pairs of integers $(x,y)$ that satisfy the system of the following equations: $$\left\{\begin{array}{l} x(y+1)=y^2-1 \\ y(x+1)=x^2-1 \end{array} \right. $$ p5. Given the following image. $ABCD$ is square, and $E$ is any point outside the square $ABCD$. Investigate whether the relationship $AE^2 + CE^2 = BE^2 +DE^2$ holds in the picture below. [img]https://cdn.artofproblemsolving.com/attachments/2/5/a339b0e4df8407f97a4df9d7e1aa47283553c1.png[/img]

Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula if \[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \] Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$

$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

A zerg player can produce one zergling every minute and a protoss player can produce one zealot every $2.1$ minutes. Both players begin building their respective units immediately from the beginning of the game. In a ght, a zergling army overpowers a zealot army if the ratio of zerglings to zealots is more than $3$. What is the total amount of time (in minutes) during the game such that at that time the zergling army would overpower the zealot army?

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the real numbers $x, y, z$ be such that $x + y + z = 0$. Prove that $$6(x^3 + y^3 + z^3)^2 \le (x^2 + y^2 + z^2)^3.$$

Let $a_1,...,a_n$ be positive real numbers such that $\sum_{i=1}^n a_i =\prod_{i=1}^n a_i $ , and let $b_1,...,b_n$ be positive real numbers such that $a_i \le b_i$ for all $i$. Prove that $\sum_{i=1}^n b_i \le\prod_{i=1}^n b_i $

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

For any positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$ As usual, [x,y] denotes the least common multiply of $x,y$ [I]Proposed by A. Golovanov[/i]

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

Ivan and Kolya play a game, Ivan starts. Initially, the polynomial $x-1$ is written of the blackboard. On one move, the player deletes the current polynomial $f(x)$ and replaces it with $ax^{n+1}-f(-x)-2$, where $\deg(f)=n$ and $a$ is a real root of $f$. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?

Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$. 1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$. 2) Find the minimum possible value of $a_0+a_1+...+a_n$.

Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Find the minimum value of \[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\] where $x,y,z>1$ are reals.