Which of the following sets of $ x$-values satisfy the inequality $ 2x^2 \plus{} x < 6?$ $ \textbf{(A)}\ \minus{} 2 < x < \frac{3}{2} \qquad \textbf{(B)}\ x > \frac32 \text{ or }x < \minus{} 2 \qquad \textbf{(C)}\ x < \frac32 \qquad \textbf{(D)}\ \frac32 < x < 2 \qquad \textbf{(E)}\ x < \minus{} 2$

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

A frog starts a journey at $(6,9).$ A skip is the act of traveling a positive integer number of units straight south or a positive integer number of units straight west. A jump is the act of traveling one unit straight west. A hop consists of any skip followed by a jump. How many different sequences of hops can the frog take so that the frog's final destination is $(0,0)$? [i]Proposed by Jack Ma[/i]

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$ [i]Proposed by Ali Zamani [/i]

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

For all $x,y,z>0$ satisfying $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z$, prove that $$\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1$$

Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A. [asy] defaultpen(linewidth(0.7)); draw(Circle(origin, 5)); draw(Circle((5,0), 5)); label("$A$", (0,5), N); label("$B$", (5,5), N); label("$1001$", (2.5, -0.5), N);[/asy] $ \textbf{(A)}\: 503\qquad \textbf{(B)}\: 1006\qquad \textbf{(C)}\: 1504\qquad \textbf{(D)}\: 1507\qquad \textbf{(E)}\: 1510\qquad $

Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 7 \qquad\textbf{d)}\ \text{Infinitely many} \qquad\textbf{e)}\ \text{None of above} $

Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

Jordan has an infinite geometric series of positive reals whose sum is equal to $2\sqrt2 + 2$. It turns out that if Jordan squares each term of his geometric series and adds up the resulting numbers, he get a sum equal to $4$. If Jordan decides to take the fourth power of each term of his original geometric series and add up the resulting numbers, what sum will he get?

There are $14$ students with different heights. At each step, two adjacent students will be swapped. Whatever the first arrangement is, in at least how many steps the students can be lined up? $\textbf{(A)}\ 42 \qquad\textbf{(B)}\ 43 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 52 \qquad\textbf{(E)}\ \text{None}$

A sequence $\{a_n\}_{n\geq 1}$ of positive reals is defined by the rule $a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2$ for integers $n>2$ together with the initial values $a_1=8$ and $a_2=64$ and $a_3=1024$. Compute \[\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}\]

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.

A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.

Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}=\triangle{KMN}.$

Positive reals $p$ and $q$ are such that the graph of $y = x^2 - 2px + q$ does not intersect the $x$-axis. Find $q$ if there is a unique pair of points $A, B$ on the graph with $AB$ parallel to the $x$-axis and $\angle AOB = \frac{\pi}{2}$, where $O$ is the origin.

Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.

Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.

Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers. (Day 1, 3rd problem author: Michal Rolínek)

Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$. [i]Janez Šter[/i]