Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Compute $\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)$.

Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

Find all non decreasing functions or non increasing function $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(f(x)))+y$$ .

Let \( f(n) \) be a polynomial with integer coefficients. Prove that if \( f(-1) \), \( f(0) \), and \( f(1) \) are not divisible by 3, then \( f(n) \neq 0 \) for all integers \( n \).

Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$, a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ with $\alpha_n \in \{0,1\}^{p(n)}$, and a deterministic polynomial time Turing Machine $M$ such that for every $x \in \{0,1\}^n$ $$x \in L \Leftrightarrow M(x, \alpha_n) = 1$$ Let us call $\alpha_n$ to be the [i]angel string [/i]for all $x$ of the length $n$. Note that the [i]angel string[/i] is $\textbf{not}$ similar to a [i]witness[/i] or [i]certificate [/i]as used in the definition of $\textbf{NP}$ For example, all unary languages, even $UHALT$ which is undecidable, are in $\textbf{P}_{angel}$ because the \textit{angel string} can simply be a single bit that tells us if the given unary string is in $UHALT$ or not. \\\\ A set $S \subseteq \Sigma^*$ is said to be [b]sparse[/b] if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ such that for each $n \in \mathbb{N}$, the number of strings of length $n$ in $S$ is bounded by $p(n)$. In other words, $|S^{=n}| \leq p(n)$, where $S^{=n} \subseteq S$ contains all the strings in $S$ that are of length $n$. [list=1] [*] Given $k \in \mathbb{N}$ sparse sets $S_1, S_2 \ldots S_k$, show that there exists a sparse set $S$ and a deterministic polynomial time TM $M$ with oracle access to $S$ such that given an input $\langle x,i \rangle$ the TM $M$ will accept it if and only if $x \in S_i$. \\Define the set $S$ (note that it need not be computable), and give the description of $M$ with oracle $S$. \\Note that a TM $M$ with oracle access to $S$ can query whether $s \in S$ and get the correct answer in return in constant time. [/*] [*] Let us define a variant of $\textbf{P}_{angel}$ called $\textbf{P}_{bad-angel}$ with a constraint that there should exists a polynomial time algorithm that can [b]compute[/b] the angel string for any length $n \in \mathbb{N}$. In other words, there is a poly-time algorithm $A$ such that $\alpha_n = A(n)$. \\Is $\textbf{P} =\textbf{P}_{bad-angel}$? Is $\textbf{NP}=\textbf{P}_{bad-angel}$? Justify. [/*] [*] Let the language $L \in$ $\textbf{P}_{angel}$. Show that there exists a sparse set $S_L$ and a deterministic polynomial time TM $M$ with oracle access to $S_L$ that can decide the language $L$. [/*]

The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.

For a given number $ n $, let us denote by $ p_n $ the probability that when randomly selecting a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ (the selection of each pair is equally probable) the number $\binom{m}{k}$ will be even. Calculate $ \lim_{n\to \infty} p_n $.

The rails on a railroad are $ 30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in: $ \textbf{(A)}\ 20\text{ seconds} \qquad\textbf{(B)}\ 2\text{ minutes} \qquad\textbf{(C)}\ 1\frac{1}{2}\text{ minutes} \qquad\textbf{(D)}\ 5\text{ minutes}\\ \textbf{(E)}\ \text{none of these}$

$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked? [i]Proposed by Andrew Wu[/i]

Let $\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}$. Prove that \[\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.\]

Let $ABC$ be an acute and scalene triangle. Points $D$ and $E$ are in the interior of the triangle so that $<$ $DAB$ $=$ $<$ $DCB$, $<$ $DAC$ $=$ $<$ $DBC$, $<$ $EAB$ $=$ $<$ $EBC$ and $<$ $EAC$ $=$ $<$ $ECB$. Prove that the triangle $ADE$ is a right triangle.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$. $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$. $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and $ \left[AC\right]$, respectively. If $ M$ and $ N$ are the foot of perpendiculars from $ S$ and $ R$ to $ PT$, then find $ \left|MN\right|$. $\textbf{(A)}\ \frac {9\sqrt {13} }{26} \qquad\textbf{(B)}\ \frac {12 \minus{} 2\sqrt {13} }{13} \qquad\textbf{(C)}\ \frac {5\sqrt {13} \plus{} 20}{13} \qquad\textbf{(D)}\ 15\sqrt {3} \qquad\textbf{(E)}\ \frac {10\sqrt {13} }{13}$