For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.

Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$

The integers $1, 2, \dots , 9$ are written on individual slips of paper and all are put into a bag. Ade chooses a slip at random, notes the integer on it, and replaces it in the bag. Bala then picks a slip at random and notes the integer written on it. Chioma then adds up Ade's and Bala's numbers. What is the probability that the unit's digit of this sum is less that $5$?

A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice? [img]https://cdn.artofproblemsolving.com/attachments/e/c/c817d92ead6cfb3868f9cb526fb4e1fd7ffe4d.png[/img]

The vertices of a regular $2012$-gon are labelled with the numbers $1$ through $2012$ in some order. Call a vertex a peak if its label is larger than the label of its two neighbours, and a valley if its label is smaller than the label of its two neighbours. Show that the total number of peaks is equal to the total number of valleys.

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

Draw on the coordinate plane a set of points $M(a, b)$ such that the equation $x^4+ax+b=0$ has a unique root satisfying the condition $0 \le x \le 1$.

Let $\triangle MBT$ be a triangle with $\overline{MB} = 4$ and $\overline{MT} = 7$. Furthermore, let circle $\omega$ be a circle with center $O$ which is tangent to $\overline{MB}$ at $B$ and $\overline{MT}$ at some point on segment $\overline{MT}$. Given $\overline{OM} = 6$ and $\omega$ intersects $ \overline{BT}$ at $I \neq B$, find the length of $\overline{TI}$. [i]Proposed by Chad Yu[/i]

An $a\times b\times c$ rectangular box is built from $a\cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1\times b \times c$ parallel to the $(b\times c)$-faces of the box contains exactly $9$ red cubes, exactly 12 green cubes, and some yellow cubes. Each of the $b$ layers of size $a\times 1 \times c$ parallel to the $(a\times c)$-faces of the box contains exactly 20 green cubes, exactly 25 yellow cubes, and some red cubes. Find the smallest possible volume of the box.

Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$. [i]A. Csaszar[/i]

Suppose that there is a set of $2016$ positive numbers, such that both their sum, and the sum of their reciprocals, are equal to $2017$. Let $x$ be one of those numbers. Find the maximum possible value of $x +\frac{1}{x}$. .

The product \[\left(1 - \frac{1}{2^{2}}\right)\left(1 - \frac{1}{3^{2}}\right)\ldots\left(1 - \frac{1}{9^{2}}\right)\left(1 - \frac{1}{10^{2}}\right)\] equals $ \textbf{(A)}\ \frac{5}{12}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{11}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{7}{10} $

Let $p = 2017$ be a prime. Given a positive integer $n$, let $T$ be the set of all $n\times n$ matrices with entries in $\mathbb{Z}/p\mathbb{Z}$. A function $f:T\rightarrow \mathbb{Z}/p\mathbb{Z}$ is called an $n$-[i]determinant[/i] if for every pair $1\le i, j\le n$ with $i\not= j$, \[f(A) = f(A'),\] where $A'$ is the matrix obtained by adding the $j$th row to the $i$th row. Let $a_n$ be the number of $n$-determinants. Over all $n\ge 1$, how many distinct remainders of $a_n$ are possible when divided by $\dfrac{(p^p - 1)(p^{p - 1} - 1)}{p - 1}$? [i]Proposed by Ashwin Sah[/i]

Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface \[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\] is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system \[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\] of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.

There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

On a circle there are $n$ points such that every point has a distinct number. Determine the number of ways of choosing $k$ points such that for any point there are at least 3 points between this point and the nearest point. (clockwise) ($n,k\geq 2$)

Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.

For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ . Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.