A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.

A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.

Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?

Sets $A, B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A\cap B| = 16$, $|A\cap C| = 51$, $|B\cap C| = 19$. Compute the number of possible values of$ |A \cap B \cap C|$.

A number of $n$ lamps ($n\ge 3$) are put at $n$ vertices of a regular $n$-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?

How many positive factors does $23,232$ have? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }42$

Consider the areas of the four triangles obtained by drawing the diagonals $AC$ and $BD$ of a trapezium $ABCD$. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.

Denote by $\lambda (H)$ the Lebesgue outer measure of $H\subseteq \left[ 0,1\right]$. The horizontal and vertical sections of the set $A\subseteq [0, 1]\times [ 0, 1]$ are denoted by $A^y$ and $A_x$ respectively; that is, $A^y=\{ x\in [ 0, 1] \colon (x, y) \in A\}$ and $A_x=\{ y\in [ 0, 1]\colon (x,y)\in A\}$ for all $x,y\in [0,1]$. (a) Is there a decomposition $A\cup B$ of the unit square $[0,1]\times [0,1]$ such that $A^y$ is the union of finitely many segments of total length less than $\frac12$ and $\lambda (B_x)\le \frac12$ for all $x, y\in [0,1]$? (b) Is there a decomposition $A\cup B$ of the unit square $[0,1] \times [0,1]$ such that $A^y$ is the union of finitely many segments of total length not greater than $\frac12$ and $\lambda (B_x)<\frac12$ for all $x,y\in [0,1]$?

Positive real numbers $x$ and $y$ satisfy $y^3 = x^2$ and $(y-x)^2 = 4y^2$. What is $x+y$? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42$

Solve the equation $a^3+b^3+c^3=2001$ in positive integers. [i]Mircea Becheanu, Romania[/i]

Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively. $M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$. Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$. Show that: $DK=DL$

In triangle $ABC$, $\measuredangle C=\theta$ and $\measuredangle B=2\theta$, where $0^{\circ} <\theta < 60^{\circ}$. The circle with center $A$ and radius $AB$ intersects $AC$ at $D$ and intersects $BC$, extended if necessary, at $B$ and at $E$ ($E$ may coincide with $B$). Then $EC=AD$ $ \textbf{(A)}\ \text{for no values of}\ \theta \qquad\textbf{(B)}\ \text{only if}\ \theta=45^{\circ} \qquad\textbf{(C)}\ \text{only if}\ 0^{\circ} < \theta \le 45^{\circ} \\ \qquad\textbf{(D)}\ \text{only if}\ 45^{\circ} \le \theta < 60^{\circ} \qquad\textbf{(E)}\ \text{for all}\ \theta \ \text{such that}\ 0^{\circ} <\theta < 60^{\circ} $

Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to: $\textbf{(A)}\ -F \qquad \textbf{(B)}\ F\qquad \textbf{(C)}\ 3F \qquad \textbf{(D)}\ F^3 \qquad \textbf{(E)}\ F^3-F$

A postman wants to divide $n$ packages with weights $1, 2, 3, 4, n$ into three groups of exactly the same weight. Can he do this if (a) $n = 2011$ ? (b) $n = 2012$ ?

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

Determine all positive integers $n$ less than $2024$ such that for all positive integers $x$, the greatest common divisor of $9x + 1$ and $nx+1$ is $1$.

$\arcsin(\sin 2000^{\circ})=$________.

Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]

Let $(G,\cdot)$ be a group and let $F$ be the set of elements in the group $G$ of finite order. Prove that if $F$ is finite, then there exists a positive integer $n$ such that for all $x\in G$ and for all $y\in F$, we have \[ x^n y = yx^n. \]

Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.

If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]