Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.

In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

Show, for all positive integers $n = 1,2 , \dots ,$ that $14$ divides $ 3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }$.

Rectangle $ABCD$ is folded in half so that the vertices $D$ and $B$ coincide, creating the crease $\overline{EF}$, with $E$ on $\overline{AD}$ and $F$ on $\overline{BC}$. Let $O$ be the midpoint of $\overline{EF}$. If triangles $DOC$ and $DCF$ are congruent, what is the ratio $BC : CD$?

If $a+b=1$, $ \in \mathbb{R}$ and $ab \ne 0$, prove that $$\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{2(ab-2)}{a^2b^2+3}$$

Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.

Consider real numbers $a, b ,c \ge0$ with $a+b+c=2$. Prove that: $\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}$

Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$: [list] [*] $f(n) > 0$. [*] $f(n)$ divides $n^{f(n)} - 1$. [/list]

The diagram below shows a square of area $36$ separated into two rectangles and a smaller square. One of the rectangles has an area of $12$. What is the smallest rectangle's area? [asy] size(70); draw((0,0)--(2,0)--(2,6)--(0,6)--cycle); draw((2,2)--(6,2)--(6,6)--(2,6)--cycle); draw((2,2)--(6,2)--(6,0)--(2,0)--cycle); label("$12$",(1,3)); label("$?$",(4,4)); label("$?$",(4,1)); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }\text{Not Enough Information}$

$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.

Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$: \[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\] For which $k$ does there exist a $k$-tastrophic function?

Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute $$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$ [i]2017 CCA Math Bonanza Individual Round #12[/i]

Triangle $\vartriangle ABC$ has side lengths $AB = 2$, $CA = 3$ and $BC = 4$. Compute the radius of the circle centered on $BC$ that is tangent to both $AB$ and $AC$.

Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?

Let $f:[0,\infty)\to\mathbb R$ be a differentiable function with $|f(x)|\le M$ and $f(x)f'(x)\ge\cos x$ for $x\in[0,\infty)$, where $M>0$. Prove that $f(x)$ does not have a limit as $x\to\infty$.

My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team? $\textbf{(A) }2/3\hspace{14em}\textbf{(B) }1\hspace{14.8em}\textbf{(C) }3/2$ $\textbf{(D) }8/5\hspace{14em}\textbf{(E) }5/8\hspace{14em}\textbf{(F) }2$ $\textbf{(G) }0\hspace{14.9em}\textbf{(H) }5/2\hspace{14em}\textbf{(I) }2/5$ $\textbf{(J) }3/4\hspace{14em}\,\textbf{(K) }4/3\hspace{13.9em}\textbf{(L) }2007$

Find all pairs $(x, y)$ of positive integers such that $(x + y)(x^2 + 9y)$ is the cube of a prime number.

Solve the equation $x^2- 10[x] + 9 = 0$. ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).

Let $N>s$ be positive integers. Electricity park has a number of buildings; exactly $N$ of them are power plants, and another one of them is the headquarter. Some pairs of buildings have one-way power cables between them, satisfying: (i) The cables connected to a power plant will only send the power out of the plant. (ii) For each non-headquarter building, there is a unique sequence of cables that can transport the power from that building to the headquarter. A building is [b]$s$-electrifed[/b] if, after removing any one cable from the park, the building can still receive power from at least $s$ different power plants. Find the maximum possible number of $s$-electrifed buildings. [i]Note: There seems to be confusion about whether a power plant is $1$-electrified. For the sake of simplicity let's say that any power plant is not $s$-electrified for any $s\geq 1$.[/i] [i]Proposed by usjl[/i]

Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.

Consider the iterated sequence (1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$, where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).

The greatest common divisor of $n$ and $180$ is $12$. The least common multiple of $n$ and $180$ is $720$. Find $n$.

Let $a$ and $b$ be positive real numbers satisfying $$a -12b = 11 -\frac{100}{a} \,\,\,\,and \,\,\,\, a -\frac{12}{b}= 4 -\frac{100}{a}.$$ Then $a + b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all ordered triples of non-negative integers $(a,b,c)$ such that $a^2+2b+c$, $b^2+2c+a$, and $c^2+2a+b$ are all perfect squares. [i]Proposed by Matthew Babbitt[/i]