Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$? $\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$

Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ?

Find all positive integers $n$ that satisfy the following inequalities: $$-46 \leq \frac{2023}{46-n} \leq 46-n$$

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Triangle $ABC$ is right angled at $C$. Lines $AM$ and $BN$ are internal angle bisectors. $AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively. Prove that the line which passes through the midpoints of segments $QN$ and $PM$ is parallel to $AB$.

Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.

Given is a convex quadrilateral $ABCD$ with $\angle BAD = \angle BCD$ and $\angle ABC < \angle ADC$. Point $M$ is the midpoint of segment $AC$. Prove that there exist points $X$ and $Y$ on the segments $AB$ and $BC$, respectively, such that $XY \perp BD, MX = MY$ and $\angle XMY = \angle ADC - \angle ABC$. [i]Proposed by Mykhailo Shtandenko[/i]

Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

Prove that if $a,b,c, d \in [1,2]$, then $$\frac{a + b}{b + c}+\frac{c + d}{d + a}\le 4 \frac{a + c}{b + d}$$ When does the equality hold?

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

Compute all prime numbers $p$ such that $8p+1$ is a perfect square.

Let $p$ and $q$ be integers. If $k^2+pk+q>0$ for every integer $k$, show that $x^2+px+q>0$ for every real number $x$.

In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.

A man has $ \$10,000$ to invest. He invests $ \$4000$ at $ 5\%$ and $ \$3500$ at $ 4\%$. In order to have a yearly income of $ \$500$, he must invest the remainder at: $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 6.1\% \qquad\textbf{(C)}\ 6.2\% \qquad\textbf{(D)}\ 6.3\% \qquad\textbf{(E)}\ 6.4\%$

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number? (D. Fomin, Leningrad)

If ri are integers such that $0 \le r_i < 31$ and $r_i$ satisfies the polynomial $x^4 + x^3 + x^2 + x \equiv 30$ (mod $31$), find $$\sum^4_{i=1}(r^2_i + 1)^{-1} \,\,\,\, (mod \,\,\,\, 31)$$ where $x^{-1}$ is the modulo inverse of $x$, that is, it is the unique integer $y$ such that $0 < y < 31$ and $xy -1$ is divisible by $31$.

$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

If $n=\overline{a_1a_2\cdots a_{k-1}a_k}$, be $s(n)$ such that . If $k$ is even, $s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}$ . If $k$ is odd, $s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}$ For example $s(123)=1+23=24$ and $s(2021)=20+21=41$ Be $n$ is $digital$ if $s(n)$ is a divisor of $n$. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is $digital$.

Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.