$F$* is the multiplicative group of the field $F$. $F$* is of finitely generated. Is it true that $F$* is cyclic? Additional question: (wasn’t at the olympiad) $K$* is the multiplicative group of the field $K$. $L \subseteq $$K$* is a finitely generated subgroup. Is it true that $L$ is cyclic?

An acute isosceles triangle, $ABC$ is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC=\angle ACB=2\angle D$ and $x$ is the radian measure of $\angle A$, then $x=$ [asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=180/7; pair D=origin, B=dir(3x), C=dir(4x), A=intersectionpoint(C--C+dir(2x), B--B+dir(5x)), O=circumcenter(A,B,C); markscalefactor=0.015; draw(B--C--D--B--A--C^^Circle(O, abs(O-C))^^anglemark(C,A,B)); dot(A^^B^^C^^D); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$x$", A+0.1*dir(270), S);[/asy] $\text{(A)} \ \frac37\pi \qquad \text{(B)} \ \frac49\pi \qquad \text{(C)} \ \frac5{11}\pi \qquad \text{(D)} \ \frac6{13}\pi \qquad \text{(E)} \ \frac7{15}\pi$

The number of digits in the number $N=2^{12}\times 5^8$ is $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad \textbf{(E) }20$

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

Consider a rectangular sheet of paper $ABCD$ such that the lengths of $AB$ and $AD$ are respectively $7$ and $3$ centimetres. Suppose that $B'$ and $D'$ are two points on $AB$ and $AD$ respectively such that if the paper is folded along $B'D'$ then $A$ falls on $A'$ on the side $DC$. Determine the maximum possible area of the triangle $AB'D'$.

There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Let $ABCD$ be a square with diagonals $AC$ and $BD$, and $P$ a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.

Given a string of at least one character in which each character is either A or B, Kathryn is allowed to make these moves: [list] [*] she can choose an appearance of A, erase it, and replace it with BB, or [*] she can choose an appearance of B, erase it, and replace it with AA. [/list] Kathryn starts with the string A. Let $a_n$ be the number of strings of length $n$ that Kathryn can reach using a sequence of zero or more moves. (For example, $a_1=1$, as the only string of length 1 that Kathryn can reach is A.) Then $\sum_{n=1}^{\infty} \frac{a_n}{5^n} = \frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Luke Robitaille[/i]

Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? $\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$

On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.

For $ n \in \mathbb{N}$, let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$. Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$

For positive real $x$, let $$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of $$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$

We consider the set of expressions that can be written with real numbers, $\pm$, $+$, $\times$, and parenthesis, such that if each $\pm$ is independently replaced with either $+$ or $-$, we are left with a valid arithmetic expression. For example, this includes: \[0\pm 1, 1 \pm 2, 1+2\times (1+2\pm 3), (1 \pm 2) \times (3 \pm 4).\] We define the [i]range[/i] of an expression of this form to be the set of all of the possible values when replacing each $\pm$ with either a $+$ or a $-$. For example, [list] [*] $1 \pm 2$ has range $\{-1,3\}$, since $1-2=-1$ and $1+2=3$. [*] $(1 \pm 1) \times (1 \pm 1)$ has range $\{0,4\}$, since $(1-1)(1-1)=(1-1)(1+1)=(1+1)(1-1)=0$ and $(1+1)(1+1)=4.$ [*] $(1 \pm 2)(3\pm 4)$ has range $\{-7,-3,1,21\}$, since $(1-2)(3+4)=-7$, $(1+2)(3-4)=-3$, $(1-2)(3-4)=1$, and $(1+2)(3+4)=21$. [/list] We will prove that every finite nonempty set of real numbers is the range of some expression of this form. Call a nonempty set of real numbers [i]good[/i] if it is the range of some expression of this form. (a) For each of the following sets, find an expression with a range equal to the given set. You do not need to justify the expression. [list=i] [*] $\{1\}$ [*] $\{1,3\}$ [*] $\{-1,0,1\}$ [/list] (b) Prove that if $S$ and $T$ are good sets, the product set $S \cdot T = \{ xy \mid x \in S, y \in T \}$ (the set of product of elements of $S$ with elements of $T$) is good. (c) Prove that if a set $S$ not containing $0$ is good, the set $S \cup \{ 0 \}$ (obtained upon adding $0$ to $S$) is good. (d) Prove that every finite nonempty set of real numbers is good.

Let $n\geqslant 3$ be a positive integer. A circular necklace is called [i]fun[/i] if it has $n{}$ black beads and $n{}$ white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. Prove that it is possible to change any fun necklace into any other fun necklace using at most $(n-1)$ moves. [i]Note:[/i] Necklaces related by rotations or reflections are considered to be the same. [i]Proposed by Dylan Toh[/i]

For non-coplanar points are given in space. A plane $\pi$ is called [i]equalizing [/i] if all four points have the same distance from $\pi$. Find the number of equilizing planes.

Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.

Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops? [i]Proposed by Harry Kim[/i]

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Anja and Bernd take turns in removing stones from a heap, initially consisting of $n$ stones ($n \ge 2$). Anja begins, removing at least one but not all the stones. Afterwards, in each turn the player has to remove at least one stone and at most as many stones as removed in the preceding move. The player removing the last stone wins. Depending on the value of $n$, which player can ensure a win?

Square $S_1$ is inscribed inside circle $C_1$, which is inscribed inside square $S_2$, which is inscribed inside circle $C_2$, which is inscribed inside square $S_3$, which is inscribed inside circle $C_3$, which is inscribed inside square $S_4$. [center]<see attached>[/center] Let $a$ be the side length of $S_4$, and let $b$ be the side length of $S_1$. What is $\tfrac{a}{b}$?

Find the value of $\frac{2014^3-2013^3-1}{2013\times 2014}$. $ \textbf{(A) }3\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }11 $

Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$