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.

The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?

Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.

Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$, \[n^n\le g(2n)\le (2n)^{2n}\]

Alice guessed that there were $5$ skittles in a jar, while Kayla guessed $19$. It turns out that they were equally close to the true number of skittles. How many skittles are in the jar? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

The archipelago Barrantes - $n$ is a group of islands connected by bridges as follows: there are a main island (Humberto), in the first step I place an island below Humberto and one above from Humberto and I connect these 2 islands to Humberto. I put $2$ islands to the left of these $2$ new islands and I connect them with a bridge to the island that they have on their right. In the second step I take the last $2$ islands and I apply the same process that I applied to Humberto. In the third step I apply the same process to the $4$ new islands. We repeat this step n times we reflect the archipelago that we have on a vertical line to the right of Humberto. We connect Humberto with his reflection and so we have the archipelago Barrantes -$n$. However, the archipelago Barrantes -$n$ exists on a small planet cylindrical, so that the islands to the left of the archipelago are in fact the islands that are connected to the islands on the right. The figure shows the Barrantes archipelago -$2$, The islands at the edges are still numbered to show how the archipelago connects around the cylindrical world, the island numbered $1$ on the left is the same as the island numbered $1$ on the right. [img]https://cdn.artofproblemsolving.com/attachments/e/c/803d95ce742c2739729fdb4d74af59d4d0652f.png[/img] One day two bands of pirates arrive at the archipelago Barrantes - $n$: The pirates Black Beard and the Straw Hat Pirates. Blackbeard proposes a game to Straw Hat: The first player conquers an island, the next player must conquer an island connected to the island that was conquered in the previous turn (clearly not conquered on a previous shift). The one who cannot conquer any island in his turn loses. Straw Hat decides to give the first turn to Blackbeard. Prove that Straw Hat has a winning strategy for every $n$.