Coin $ A$ is flipped three times and coin $ B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? $ \textbf{(A)}\ \frac {19}{128}\qquad \textbf{(B)}\ \frac {23}{128}\qquad \textbf{(C)}\ \frac {1}{4}\qquad \textbf{(D)}\ \frac {35}{128}\qquad \textbf{(E)}\ \frac {1}{2}$

For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "[i]$(P, Q)$-travel route[/i]" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions. [list] [*] $T_0=P,\ T_s=Q$ [*] $T_0, T_1, \ldots, T_s$ are all distinct. [*] There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$. [/list] Prove that there exist two airports $P, Q$ such that there is no or exactly one [i]$(P, Q)$-travel route[/i]. [hide=Graph Wording]Consider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$. [/hide]

Solve the equation $$(12x-1)(6x-1)(4x-1)(3x -1) = 5.$$

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

Triangle $ABC$ has incircle centered at $I.$ Define $M$ and $N$ the midpoints of $BC$ and $CA,$ respectively. Extend $BI$ and $MN$ to meet at a point $K.$ The circumcircle of $\triangle BKC$ intersects the incircle at two points $D$ and $G,$ where $D$ is closer to $AB$ than $G.$ Line $BK$ intersects the incircle at two points $E$ and $F,$ where $FK<EK.$ Let $H$ be $DC \cap BK$. Given that $BD=3$ and $DF=4,$ compute $\tfrac{BE}{EF} \cdot \tfrac{BH}{HF}.$ [i]Tiebreaker #3[/i]

The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$? $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 4 \qquad \text{(D)} \ 8\qquad \text{(E)} \ 16$

From point $D$ of parallelogram $ABCD$ were drawn an arbitrary line $\ell_1$, intersecting the segment $AB$ and the line $BC$ at points $C_1$ and $A_1$, respectively, and an arbitrary line $\ell_2$ intersecting the segment $BC$ and the line $AB$ at the points $A_2$ and $C_2$, respectively. Find the locus of the intersection points of the circles $(A_1BC_2)$ and $(A_2BC_1)$ (other than point $B$).

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

Factor completely the expression $(a-b)^3+(b-c)^3+(c-a)^3$

Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.

[b]p1.[/b] Find the greatest integer $n$ such that $n \log_{10} 4$ does not exceed $\log_{10} 1998$. [b]p2.[/b] Rectangle $ABCD$ has sides $AB = CD = 12/5$, $BC = DA = 5$. Point $P$ is on $AD$ with $\angle BPC = 90^o$. Compute $BP + PC$. [b]p3.[/b] Compute the number of sequences of four decimal digits $(a, b, c, d)$ (each between $0$ and $9$ inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.) [b]p4.[/b] Solve for $t$, $-\pi/4 \le t \le \pi/4 $: $$\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}$$ [b]p5.[/b] Find all integers $n$ such that $n - 3$ divides $n^2 + 2$. [b]p6.[/b] Find the maximum number of bishops that can occupy an $8 \times 8$ chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.) [b]p7.[/b] Points $A, B, C$, and $D$ are on a Cartesian coordinate system with $A = (0, 1)$, $B = (1, 1)$, $C = (1,-1)$, and $D = (-1, 0)$. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$. [b]p8.[/b] Find the number of distinct real values of $x$ which satisfy $$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an equation $$x^3 + ?x^2 + ?x + ? = 0$$ First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)

Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?

Fix a triangle $ABC$. The variable point $M$ in its interior is such that $\angle MAC = \angle MBC$ and $N$ is the reflection of $M$ with respect to the midpoint of $AB$. Prove that $|AM| \cdot |BM| + |CM| \cdot |CN|$ is independent of the choice of $M$.

The two cats Fitz and Will play the following game. On a blackboard is written the expression \[ x^{100} + {\square} x^{99} + {\square} x^{98} + {\square} x^{97} + \dots + {\square } x^2 + {\square} x +1. \] Both cats take alternate turns replacing one $\square$ with a $0$ or $1$, with Fitz going first, until (after 99 turns) all the blanks have been filled. If the resulting polynomial obtained has a real root, then Will wins, otherwise Fitz wins. Determine, with proof, which player has a winning strategy.

Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$. Compute $s^4 -18s^2 - 8s$ .

Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always speak the truth or always lie. Arnaldo can ask questions for any of these people provided these questions are of the type: “In set $A$, what is the parity of people who speak to true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only be “$even$” or “$odd$”. a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and which people always lie? b) What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite?

Alice has five real numbers $a < b < c < d < e$. She takes the sum of each pair of numbers and writes down the ten sums. The three smallest sums are $32$, $36$ and $37$, while the two largest sums are $48$ and $51$. Determine $e$.

Determine all integers $n\ge 2$ with the following property: there exist nonzero real numbers $x_1, x_2, \ldots, x_n,y$ such that \[(x_1+x_2+\ldots+x_k)(x_{k+1}+x_{k+2}+\ldots+x_n)=y\] for all $k\in\{1,2,\ldots,n-1\}$.

Prove that $\frac{1}{x+y+1}-\frac{1}{(x+1)(y+1)}<\frac{1}{11}$ for all positive $x$ and $y$.

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $m,\ n$ be the given distinct positive integers. Answer the following questions. (1) Find the real number $\alpha \ (|\alpha |<1)$ such that $\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0$. (2) Find the real number $\beta$ satifying the sytem of equation $\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}$, $\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}$.

Let $n$ and $a \leq n$ be two positive integers. There's $2n$ people sitting around a circle reqularly. Two people are friend iff one of their distance in the circle is $a$(that is , $a-1$ people are between them). Find all integers $a$ in terms of $n$ st we can choose $n$ of these people , no two of them positioned in front of each other(means they're not antipodes of each other in the circle) and the total friendship between them is an odd number.