In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.

Let $-1 < x < 1.$ Show that \[ \sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}. \] Deduce that \[ \csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8. \]

[b]p4.[/b] For how many three-digit multiples of $11$ in the form $\underline{abc}$ does the quadratic $ax^2 + bx + c$ have real roots? [b]p5.[/b] William draws a triangle $\vartriangle ABC$ with $AB =\sqrt3$, $BC = 1$, and $AC = 2$ on a piece of paper and cuts out $\vartriangle ABC$. Let the angle bisector of $\angle ABC$ meet $AC$ at point $D$. He folds $\vartriangle ABD$ over $BD$. Denote the new location of point $A$ as $A'$. After William folds $\vartriangle A'CD$ over $CD$, what area of the resulting figure is covered by three layers of paper? [b]p6.[/b] Compute $(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(x_n)_{n\geq 1}$ be an increasing and unbounded sequence of positive integers such that $x_1=1$ and $x_{n+1}\leq 2x_n$ for all $n\geq 1$. Prove that every positive integer can be written as a finite sum of distinct terms of the sequence. [i]Note:[/i] Two terms $x_i$ and $x_j$ of the sequence are considered distinct if $i\neq j$.

Determine all the pairs of positive real numbers $(a, b)$ with $a < b$ such that the following series $$\sum_{k=1}^{\infty} \int_a^b\{x\}^k dx =\int_a^b\{x\} dx + \int_a^b\{x\}^2 dx + \int_a^b\{x\}^3 dx + \cdots$$ is convergent and determine its value in function of $a$ and $b$. [b]Note: [/b] $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

Prove that $$\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}} \leq\frac{3\sqrt{3}}{2}$$ for any positive real numbers $x, y,z$ such that $x+y+z = xyz.$ [url=https://artofproblemsolving.com/community/c7h236610p10925499]2008 VTRMC #1[/url] [url=http://www.math.vt.edu/people/plinnell/Vtregional/solutions.pdf]here[/url]

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x>y>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her values. Which of the following statements is necessarily correct? $ \textbf{(A)}\ \text{Her estimate is larger than }x-y\\ \textbf{(B)}\ \text{Her estimate is smaller than }x-y\\ \textbf{(C)}\ \text{Her estimate equals }x-y\\ \textbf{(D)}\ \text{Her estimate equals }y-x\\ \textbf{(E)}\ \text{Her estimate is }0 $

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

We call a set $ S$ on the real line $ \mathbb{R}$ [i]superinvariant[/i] if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0$ there exists a translation $ B,$ $ B(x) \equal{} x\plus{}b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) \equal{} B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) \equal{} A(u).$ Determine all [i]superinvariant[/i] sets.

A $k$-coloring of a graph $G$ is a coloring of its vertices using $k$ possible colors such that the end points of any edge have different colors. We say a graph $G$ is uniquely $k$-colorable if one hand it has a $k$-coloring, on the other hand there do not exist vertices $u$ and $v$ such that $u$ and $v$ have the same color in one $k$-coloring and $u$ and $v$ have different colors in another $k$-coloring. Prove that if a graph $G$ with $n$ vertices $(n \ge 3)$ is uniquely $3$-colorable, then it has at least $2n-3$ edges.

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$

Find the area of the region in the $xy$-plane satisfying $x^6-x^2+y^2 \le 0$.

The positive integers from $1$ to $n^2$ are placed arbitrarily on the squares of a chess board with dimensions $n\times n$. Prove that there are two adjacent squares (having a common vertex or a common side) such that the difference between the numbers placed on them is not less than $n + 1$. (N Sedrakyan, Yerevan)

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

Submit an integer between $0$ and $100$ inclusive as your answer to this problem. Suppose that $Q_1$ and $Q_3$ are the medians of the smallest $50\%$ and largest $50\%$ of submissions for this question. Your goal is to have your submission close to $D=Q_3-Q_1$. If you submit $N$, your score will be $2-2\sqrt{\frac{\left|N-D\right|}{\max\left\{D,100-D\right\}}}$. [i]2019 CCA Math Bonanza Lightning Round #5.4[/i]

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides.

How man y positive integers less than $50$ have an odd number of positive integer divisors? $\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11$