Find $ax^5 + by^5$ if the real numbers $a$, $b$, $x$, and $y$ satisfy the equations \begin{eqnarray*} ax + by &=& 3, \\ ax^2 + by^2 &=& 7, \\ ax^3 + by^3 &=& 16, \\ ax^4 + by^4 &=& 42. \end{eqnarray*}

Jacob flips five coins, exactly three of which land heads. What is the probability that the first two are both heads?

Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$.

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

Suppose that $n$ squares of an infinite square grid are colored grey, and the rest are colored white. At each step, a new grid of squares is obtained based on the previous one, as follows. For each location in the grid, examine that square, the square immediately above, and the square immediately to the right. If there are two or three grey squares among these three, then in the next grid, color that location grey, otherwise, color it white. Prove that after at most n steps all the squares in the grid will be white. Below is an example with $n = 4$. The first grid shows the initial configuration, and the second grid shows the configuration after one step. [img]https://cdn.artofproblemsolving.com/attachments/1/a/87f7e3892cdb45fb3529127234aae2cea08749.png[/img]

Let $F_1=F_2=1$. We define inductively $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 2$. Then the sum $$F_1+F_2+\cdots+F_{2017}$$ is [list=1] [*] Even but not divisible by 3 [*] Odd but divisible by 3 [*] Odd and leaves remainder 1 when divisible by 3 [*] None of these [/list]

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear. [i]Proposed by Holden Mui[/i]

Let $n$ be a positive integer and let $x_1$, $\ldots$, $x_n$ be positive real numbers. Show that: \[ \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ). \]

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$

Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. For example, if there are smudges on $3$ and $9,$ two possible codes are $393939$ and $993999.$ Which number of distinct digits in the access code offers the most combinations?

At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT". [i]Proposed by Alex Li[/i]

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

The difference of the roots of $ x^2 \minus{} 7x \minus{} 9 \equal{} 0$ is: $ \textbf{(A)}\ \plus{} 7 \qquad\textbf{(B)}\ \plus{} \frac {7}{2} \qquad\textbf{(C)}\ \plus{} 9 \qquad\textbf{(D)}\ 2\sqrt {85} \qquad\textbf{(E)}\ \sqrt {85}$

Given $n \geq 2$ reals $x_1 , x_2 , \dots , x_n.$ Show that $$\prod_{1\leq i < j \leq n} (x_i - x_j)^2 \leq \prod_{i=0}^{n-1} \left(\sum_{j=1}^{n} x_j^{2i}\right)$$ and find all the $(x_1 , x_2 , \dots , x_n)$ where the equality holds.

Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \ne D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$. Show that $A, X$, and $Y$ are collinear.

Given a triangle $ABC$. $D$ is a moving point on the edge $BC$. Point $E$ and Point $F$ are on the edge $AB$ and $AC$, respectively, such that $BE=CD$ and $CF=BD$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ intersects at another point $P$ other than $D$. Prove that there exists a fixed point $Q$, such that the length of $QP$ is constant.

Ten pirates find a chest filled with golden and silver coins. There are twice as many silver coins in the chest as there are golden. They divide the golden coins in such a way that the difference of the numbers of coins given to any two of the pirates is not divisible by $10.$ Prove that they cannot divide the silver coins in the same way.

For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.

Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$.

There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two people take turns removing stone, each round they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer $m$ such that LTF has a winning strategy. [i]Proposed by ltf0501[/i]

While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$ (A)$20$ (B)$15$ (C)$9$ (D)$8$

Alice and Bob are put in charge of building a bridge with their respective teams. With both teams' combined effort, the team can be finished in $6$ days. In reality, Alice's team works alone for the first $3$ days, and then, they decide to take a break. Bob's team takes over from there and works for another $4$ days. As a result, $60\%$ of the bridge is successfully constructed. How many days would it take for Alice's team alone to finish building the bridge completely from the start?