Prove: [b](a)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])=1.$$ [b](b)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])>1.$$

Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$. [i]2016 CCA Math Bonanza Lightning #3.2[/i]

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges?

Let $ABCD$ be a tetrahedron and $A'$, $B'$, $C'$ be arbitrary points on the edges $[DA]$, $[DB]$, $[DC]$, respectively. One considers the points $P_c \in [AB]$, $P_a \in [BC]$, $P_b \in [AC]$ and $P'_c \in [A'B']$, $P'_a \in [B'C']$, $P'_b \in [A'C']$ such that $$\frac{P_cA}{P_cB}= \frac{P'_cA'}{P'_cB'}=\frac{AA'}{BB'}\,\,\, , \,\,\,\frac{P_aB}{P_aC}= \frac{P'_aB'}{P'_aC'}=\frac{BB'}{CC'}\,\,\, , \,\,\, \frac{P_bC}{P_bA}= \frac{P'_bC'}{P'_bA'}=\frac{CC'}{AA'}$$ Prove that: a) the lines $AP_a,$ $BP_b$, $CP_c$ have a common point $P$ and the lines $A'P'_a$, $B'P'_b$ , $C'P'_c$ have a common point $P'$ b) $\frac{PC}{PP_c}=\frac{P'C'}{P'P'_c} $ c) if $A', B', C'$ are variable points on the edges $[DA]$, $[DB]$, $[DC]$, then the line $PP'$ is always parallel to a fixed line.

Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality $$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$

The number of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990$. Find the greatest number of apples growing on any of the six trees.

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$. Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

Let $ABC$ be a triangle with $AB < AC < BC$ and $\Gamma$ its circumcircle. Let $\omega_1$ be the circle with center $B$ and radius $AC$ and $\omega_2$ the circle with center $C$ and radius $AB$. The circles $\omega_1$ and $\omega_2$ intersect at point $E$ such that $A$ and $E$ are on opposite sides of the line $BC$. The circles $\Gamma$ and $\omega_1$ intersect at point $F$ and the circles $\Gamma$ and $\omega_2$ intersect at point $G$ such that the points $F$ and $G$ are on the same side as $E$ in relation to the line $BC$. With $K$ being the point such that $AK$ is a diameter of $\Gamma$, prove that $K$ is circumcenter of triangle $EFG$.

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values ​​of all angles of triangle $ABC$.

The positive integers $a, b, c, d$ satisfy (i) $a + b + c + d = 2023$ (ii) $2023 \text{ } | \text{ } ab - cd$ (iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$ Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.

A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.[img]https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png[/img]

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

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 $f(85)$?