A coin is flipped three times. What is the probability that there are no instances of two consecutive heads or two consecutive tails? $\textbf{(A) }\frac{1}{8}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{8}\qquad\textbf{(E) }\frac{3}{4}$

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done? $\textbf{(A) }0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions $i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct). $ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$

Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.

Find all pairs $(x,y)$ of real numbers that satisfy, \begin{align*} x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\ |xy| & \leq \frac{25}{9}. \end{align*}

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]

Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.

A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Find all real numbers $a$ for which there exist different real numbers $b, c, d$ different from $a$ such that the four tangents drawn to the curve $y = \sin (x)$ at the points $(a, \sin (a)), (b, \sin (b)), (c, \sin (c))$ and $(d, \sin (d))$ form a rectangle.

Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]

Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side. Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.

Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.

Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers. [i]Proposed by Michael Ren[/i]

A regular octahedron has side length $ 1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $ \frac {a\sqrt {b}}{c}$, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ b$ is not divisible by the square of any prime. What is $ a \plus{} b \plus{} c$? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$

In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.

Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \] for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \] for every $n \in \mathbb{N}$.

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

Let $ABC$ be an acute triangle inscribed in a circle $(O)$ that is fixed, and two of the vertices $B$, $C$ are fixed while vertex $A$ varies on the circumference of the circle. Let $I$ be the center of the incircle, and $AD$ the angle bisector. Let $K$, $L$ be the circumcenters of $CAD$, $ABD$. A line through $O$ parallel to $DL$, $DK$ intersects the line that is through $I$ perpendicular to $IB$, $IC$ at $M$, $N$ respectively. Prove that $MN$ is tangent to a fixed circle when $A$ varies on the circle $(O)$. Tran Quang Hung, Natural Science High School, National University, Hanoi

Tom, Dick and Harry started out on a $ 100$-mile journey. Tom and Harry went by automobile at the rate of $ 25$ mph, while Dick walked at the rate of $ 5$ mph. After a certain distance, Harry got off and walked on at $ 5$ mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was: $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{none of these answers}$

Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$ [hide]If possible, don't use projective geometry[/hide]

A rise of $ 600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $ 3\%$ to $ 2\%$ is approximately: $ \textbf{(A)}\ 10000 \text{ ft.} \qquad\textbf{(B)}\ 20000 \text{ ft.} \qquad\textbf{(C)}\ 30000 \text{ ft.} \qquad\textbf{(D)}\ 12000 \text{ ft.} \qquad\textbf{(E)}\ \text{none of these}$

Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$