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$.

Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?

Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$, then select $k$ numbers in $C$, calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean. If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$.

Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper? [asy]size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label("$A$",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5)); draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label("$A$",(6.5,0),NW); dot((6.5,0)); [/asy] $\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h $

If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?

In trapezoid $ABCD$ with $AB$ parallel to $CD$ show that : $\frac{|AB|^2-|BC|^2+|AC|^2}{|CD|^2-|AD|^2+|AC|^2}=\frac{|AB|}{|CD|}=\frac{|AB|^2-|AD|^2+|BD|^2}{|CD|^2-|BC|^2+|BD|^2}$

Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.

A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the respect to the horizontal axis that crosses the center. This is illustrated in the figure. The ball flies up in the air, making a parabola and, when it comes back down, it does not hit the turntable. This can happen only if $\theta>\theta_m$. Find $\theta_m$, rounded to the nearest integer degree? [asy] filldraw(circle((0,0),1),gray(0.7)); draw((0,0)--(0.81915, 0.57358)); dot((0.81915, 0.57358)); draw((0.81915, 0.57358)--(0.475006, 1.06507)); arrow((0.417649,1.14698), dir(305), 12); draw((0,0)--(1,0),dashed); label("$\theta$", (0.2, 0.2/3), fontsize(8)); label("$r$", (0.409575,0.28679), NW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

There are two very strict laws in the country of Militaria. (i) Anyone who is shorter than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $r$ from him) is freed from the military service. (ii) Anyone who is taller than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $R$ from him) is allowed to serve in the police. A nice thing is that each man $X$ may choose his own (possibly different) positive numbers $r = r(X)$ and $R = R(X)$. Can it happen that $90\%$ (or more) of the men in Militaria are free from the army and, at the same time, $90\%$ (or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.) (N Konstantinov)

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.

Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$. Find the minimal natural number $k$, satisfying the condition: for any $M$, we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$, $A$ and $f(A)$ are colored with different colors.

The first $32$ perfect squares, $1$, $4$, $9$, $16$, $25$, $\ldots$, $961$, $1024$ are combined together into one large number by appending their digits in succession, forming the number $N = 1491625\ldots9611024$. How many digits does $N$ have? $\textbf{(A) }84\qquad\textbf{(B) }85\qquad\textbf{(C) }86\qquad\textbf{(D) }87\qquad\textbf{(E) }88$

Let $K > 0$ be an integer. An integer $k \in [0,K]$ is randomly chosen. A sequence of integers is defined starting on $k$ and ending on $0$, where each nonzero term $t$ is followed by $t$ minus the largest Lucas number not exceeding $t$. The probability that $4$, $5$, or $6$ is in this sequence approaches $\tfrac{a - b \sqrt c}{d}$ for arbitrarily large $K$, where $a$, $b$, $c$, $d$, are positive integers, $\gcd(a,b,d) = 1$, and $c$ is squarefree. Find $a + b + c + d$. [i](Lucas numbers are defined as the members of the infinite integer sequence $2$, $1$, $3$, $4$, $7$, $\ldots$ where each term is the sum of the two before it.)[/i] [i]Proposed by Evan Chang[/i]

$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?

The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$

There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?

Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

Three different circles of equal radii intersect in point $Q$. The circle $C$ touches all of them. Prove that $Q$ is the center of $C$.

The number of points common to the graphs of \[(x-y+2)(3x+y-4)=0\text{ and }(x+y-2)(2x-5y+7)=0\] is: $\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{infinite}$

For positive real numbers $a, b, c$ such that $a+b+c=3$, show that: \[\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.\]