Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.

Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined. (V Proizvolov)

Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR. ($A\oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2= 12$ and $6 \oplus 7 = 110_2 + 111_2 = 001_2 = 1$.) [i]Proposed by Jacob Weiner[/i]

A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ of side length 112. Then, for each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$, forming an infinite sequence of circles tangent to sides $XY$ and $XZ$ and approaching vertex $X$. If these circles collectively have area $m\pi$, find $m$. [i]Proposed by Michael Tang[/i]

[u]Set 4[/u] [b]p10.[/b] Eve has nine letter tiles: three $C$’s, three $M$’s, and three $W$’s. If she arranges them in a random order, what is the probability that the string “$CMWMC$” appears somewhere in the arrangement? [b]p11.[/b] Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell? [b]p12.[/b] A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck. PS. You should use hide for answers.

An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that \[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1 \] for every pair of distinct nonnegative integers $ i, j$.

A number which when divided by $ 10$ leaves a remainder of $ 9$, when divided by $ 9$ leaves a remainder of $ 8$, by $ 8$ leaves a remainder of $ 7$, etc., down to where, when divided by $ 2$, it leaves a remainder of $ 1$, is: $ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 419 \qquad\textbf{(C)}\ 1259 \qquad\textbf{(D)}\ 2519 \qquad\textbf{(E)}\ \text{none of these answers}$

Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\).

Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Lewis Chen [/i]

Let circles $C_1$ and $C_2$ be internally tangent at point $P$, with $C_1$ being the smaller circle. Consider a line passing through $P$ which intersects $C_1$ at $Q$ and $C_2$ at $R$. Let the line tangent to $C_2$ at $R$ and the line perpendicular to $\overline{PR}$ passing through $Q$ intersect at a point $S$ outside both circles. Given that $SR = 5$, $RQ = 3$, and $QP = 2$, compute the radius of $C_2$.

Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?

2/1/31. Let $x, y,$ and $z$ be real numbers greater than $1$. Prove that if $x^y = y^z = z^x$, then $x = y = z$.

\[\int_0^1 x\sqrt[4]{1-x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by $$q(x) = \prod_{j=1}^{2015} p(x + j)$$. We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$ such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded? [asy] size(5cm); defaultpen(linewidth(1pt)); draw(circle((3,3),3)); filldraw(circle((5.5,3),0.5),mediumgray*0.5 + lightgray*0.5); filldraw(circle((2,3),2),mediumgray*0.5 + lightgray*0.5); filldraw(circle((1,3),1),white); filldraw(circle((3,3),1),white); add(grid(6,6,mediumgray*0.5+gray*0.5+linetype("4 4"))); filldraw(circle((4.5,4.5),0.5),mediumgray*0.5 + lightgray*0.5); filldraw(circle((4.5,1.5),0.5),mediumgray*0.5 + lightgray*0.5); [/asy]$\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac{11}{36}\qquad\textbf{(C) } \dfrac13\qquad\textbf{(D) } \dfrac{19}{36}\qquad\textbf{(E) } \dfrac59$

Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.

Let $p$ be an odd prime. Prove that the number $$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$ is divisible by $20p$.

$\lim_{n\to\infty } \sum_{1\le i\le j\le n} \frac{\ln (1+i/n)\cdot\ln (1+j/n)}{\sqrt{n^4+i^2+j^2}} $ [i]Gabriel Mîrșanu[/i] and [i]Andrei Nedelcu[/i]

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows: $$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$ where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example, $f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$. $a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$. $b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.