Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$? $ \textbf{(A)}\ \frac{65}{8} \qquad \textbf{(B)}\ \frac{25}{3} \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ \frac{75}{8} \qquad \textbf{(E)}\ \frac{85}{8}$

Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5. [i]Proposed by Matthew Lerner-Brecher[/i]

[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person. (a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable. (b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable. [b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$? [b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$. (b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions. [b]p4.[/b] Let function $f(x, y, z)$ be defined as following: $$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$ Find the minimum value and prove the result. [b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.

Is the number \[ \left( 1 + \frac12 \right) \left( 1 + \frac14 \right) \left( 1 + \frac16 \right)\cdots\left( 1 + \frac{1}{2018} \right) \] greater than, less than, or equal to $50$?

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? $ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?

Convex quadrilateral $ABCD$ has $AB=20$, $BC=CD=26$, and $\angle{ABC}=90^\circ$. Point $P$ is on $DA$ such that $\angle{PBA}=\angle{ADB}$. If $PB=20$, compute the area of $ABCD$. [i]2019 CCA Math Bonanza Individual Round #13[/i]

Let $ABC$ be a triangle inscribed in a circle $\omega$ and $\ell$ be the tangent to $\omega$ at $A$. The line through $B$ parallel to $AC$ meets $\ell$ at $P$, and the line through $C$ parallel to $AB$ meets $\ell$ at $Q$. The circumcircles of $ABP$ and $ACQ$ meet at $S\neq A$. Show that $AS$ bisects $BC$. [i]Proposed by Andrew Gu.[/i]

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

Find all possible finite sequences $\{n_0, n_1, n_2, \ldots, n_k \}$ of integers such that for each $i, i$ appears in the sequence $n_i$ times $(0 \leq i \leq k).$

Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$. ${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $

Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?

Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$. [i]2022 CCA Math Bonanza Individual Round #3[/i]

Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

Tom's age is $ T$ years, which is also the sum of the ages of his three children. His age $ N$ years ago was twice the sum of their ages then. What is $ \frac {T}{N}$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Given an $n \times n$ ($n \ge 3$) square table with one of the following unit vectors $\uparrow, \downarrow, \leftarrow, \rightarrow$ in any its cell (the vectors are parallel to the sides and the middles of them coincide with the centers of the cells). Per move a beetle creeps from one cell to another in accordance with the vector’s direction. If the beetle starts from any cell, then it comes back to this cell after some number of moves. The vectors are directed so that they do not allow the beetle to leave the table. Is it possible that the sum of all vectors at any row (except for the first one and the last one) is equal to the vector that is parallel to this row, and the sum of all vectors at any column (except for the first one and the last one) is equal to the vector that is parallel to this column ? (D. Dudko)

Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously. a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint. b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.

Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate \[\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)\]

Convex quadrilateral $ABCD$ has side-lengths $AB=7$, $BC=9$, $CD=15$, and there exists a circle, lying inside the quadrilateral and having center $I$, that is tangent to all four sides of the quadrilateral. Points $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively. It can be proven that point $I$ always lies on segment $MN$. Supposing further that $I$ is the midpoint of $MN$, the area of quadrilateral $ABCD$ may be expressed as $p\sqrt q$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p\cdot q$.

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.

Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. What possible values can $a^3 + \frac{1}{a^3}$ and $a^5 + \frac{1}{a^5}$ take?

Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.