Indicate the moment in time when for the first time after midnight the angle between the minute and hour hands will be equal to $1^o$, despite the fact that the minute hand shows an integer number of minutes.

How many different real numbers $x$ satisfy the equation $$(x^2-5)^2=16?$$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

Prove that the product of five consecutive positive integers cannot be a perfect square.

Decompose a $6\times 6$ square into unit squares and consider the $49$ vertices of these unit squares. We call a square good if its vertices are among the $49$ points and if its sides and diagonals do not lie on the gridlines of the $6\times 6$ square. [list=a] [*]Find the total number of good squares. [*]Prove that there exist two good disjoint squares such that the smallest distance between their vertices is $1/\sqrt{5}.$ [/list]

On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1+3+5+7+9+11+13=49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$th, and let $P$ be that perfect square. Find $N+P$. [i]2018 CCA Math Bonanza Lightning Round #2.3[/i]

Let $x, y, z$ be positive real numbers whose sum is $2013$. Find the maximum possible value of $$\frac{(x^2+y^2+z^2)(x^3+y^3+z^3)}{ (x^4+y^4+z^4)}.$$

There is a city with $n$ metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations $A$ and $B$. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and $A$ and $B$ are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If $n$ is odd, prove that for any integer $k$ $(0<k<n)$ there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through $k$ metro intervals.

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

Find the range value of $a$, satisfyin that $\forall x\in\mathbb{R},\theta\in\left[0,\frac{\pi}{2}\right]$, $$(x+3+2\sin\theta\cos\theta)^2+(x+a\sin\theta+a\cos\theta)^2\geq\frac{1}{8}.$$

Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?

Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

Suppose function $f:[0,\infty)\to[0,\infty)$ satisfies (1)$\forall x,y \geq 0,$ we have $f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2})$; (2)$\forall 0 \leq x \leq 1, f(x) \leq 2016$. Prove that $f(x)\leq x^2$ for all $x\geq 0$.

Paca-Vaca decided to note every day a single quadratic polynomial of the form $x^2+ax+b$, where $a$ and $b$ are positive integers, less or equal than $100$. He follows the rule that the polynomial he writes must not have any common roots with the polynomials previously noted. What is the maximum amount of days Paca-Vaca can follow this plan?

Thirty numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)

Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.

A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? [asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25 $

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

Let $ABC$ be a triangle with $\angle BAC = 60^\circ, BA = 2$, and $CA = 3$. A point $M$ is located inside $ABC$ such that $MB = 1$ and $MC = 2$. A semicircle tangent to $MB$ and $MC$ has its center $O$ on $BC$. Let $P$ be the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. If the ratio $OP/MO$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.

The numbers $2, 2, ..., 2$ are written on a blackboard (the number $2$ is repeated $n$ times). One step consists of choosing two numbers from the blackboard, denoting them as $a$ and $b$, and replacing them with $\sqrt{\frac{ab + 1}{2}}$. $(a)$ If $x$ is the number left on the blackboard after $n - 1$ applications of the above operation, prove that $x \ge \sqrt{\frac{n + 3}{n}}$. $(b)$ Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}$ [i]Proposed by Ali Zamani, Iran[/i]

Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.

The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$.

For each integer $n \ge 2$, find all integer solutions of the following system of equations: \[x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}\] \[x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}\] \[\vdots\] \[x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}\]

Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.