The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$? [asy]//Choice A size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125; } real g(real x) { return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(A)}$", (-5,4.5)); [/asy] [asy]//Choice B size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2; } real g(real x) { return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(B)}$", (-5,4.5)); [/asy] [asy]//Choice C size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.21875 x^2+0.28125 x+0.5; } real g(real x) { return -0.375 x^2-0.75 x+0.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(C)}$", (-5,4.5)); [/asy] [asy]//Choice D size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875; } real g(real x) { return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625; } real z=3.14; draw(graph(f,-z, z), heavygray); draw(graph(g,-z, z), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(D)}$", (-5,4.5)); [/asy] [asy]//Choice E size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598; } real g(real x) { return -0.166667 x^3+0.125 x^2+0.479167 x-0.375; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(E)}$", (-5,4.5)); [/asy]

Let $\Gamma$ and $\Omega$ be circles that are internally tangent at a point $P$ such that $\Gamma$ is contained entirely in $\Omega$. Let $A,B$ be points on $\Omega$ such that the lines $PB$ and $PA$ intersect the circle $\Gamma$ at $Y$ and $X$ respectively, where $X,Y\neq P$. Let $O_1$ be the circle with diameter $AB$ and $O_2$ be the circle with diameter $XY$. Let $F$ be the foot of $Y$ on $XP$. Let $T$ and $M$ be points on $O_1$ and $O_2$ respectively such that $TM$ is a common tangent to $O_1$ and $O_2$. Let $H$ be the orthocenter of $\triangle ABP$. Given that $PF=12$, $FX=15$, $TM=18$, $PB=50$, find the length of $AH$.

Find all continuous functions $f : R \rightarrow R$ such, that $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$ for any real numbers $x$ and $y$

The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$.

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find an expression for $\beta$ in terms of $k$. $ \textbf{(A)}\ 1+k^2$ $ \textbf{(B)}\ \sqrt{1+k^2}$ $ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$ $ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$ $ \textbf{(E)}\ \text{none of the above}$

[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$. [b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.

Let $ABC$ be a triangle with bisectors $BE$ and $CF$ meet at $I$. Let $D$ be the projection of $I$ on the $BC$. Let M and $N$ be the orthocenters of triangles $AIF$ and $AIE$, respectively. Lines $EM$ and $FN$ meet at $P.$ Let $X$ be the midpoint of $BC$. Let $Y$ be the point lying on the line $AD$ such that $XY \perp IP$. Prove that line $AI$ bisects the segment $XY$. [i]Proposed by Tran Quang Hung - Vietnam[/i]

Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove: $$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$

$99$ different points $P_1, P_2, ..., P_{99}$ are marked on circle $O$. For each $P_i$, define $n_i$ as the number of marked points you encounter starting from $P_i$ to its antipode, moving clockwise. Prove the following inequality. $$n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900$$

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Find all positive integers $n$ for which there exists a set of exactly $n$ distinct positive integers, none of which exceed $n^2$, whose reciprocals add up to $1$.

For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$. Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$ [i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]

In the right triangle $ABC$ shown, $E$ and $D$ are the trisection points of the hypotenuse $AB$. If $CD=7$ and $CE=6$, what is the length of the hypotenuse $AB$? Express your answer in simplest radical form. [asy] pair A, B, C, D, E; A=(0,2.9); B=(2.1,0); C=origin; D=2/3*A+1/3*B; E=1/3*A+2/3*B; draw(A--B--C--cycle); draw(C--D); draw(C--E); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,NE); [/asy]

Evaluate $$1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.$$ $\text{(A) }\frac{\sqrt{3}}{2}\qquad\text{(B) }\frac{1+\sqrt{5}}{2}\qquad\text{(C) }\frac{2+\sqrt{3}}{2}\qquad\text{(D) }\frac{3+\sqrt{5}}{2}\qquad\text{(E) }\frac{3+\sqrt{13}}{2}$

For a set $M$ of $n\geq 3$ points in the plane we define a [i]path[/i] to be a polyline $A_1A_2\dots A_n$, where $M=\{A_1,A_2,\dots ,A_n\}$ and define its length to be $A_1A_2+A_2A_3+\dots +A_{n-1}A_n$. We call $M$ [i]path-unique[/i] if any two distinct paths have different lengths and [i]segment-unique[/i] if any two nondegenerate segments with their ends among the points in $M$ have different lengths. Determine the positive integers $n\geq 3$ such that: a) any segment-unique set $M$ of $n$ points in the plane is path-unique; b) any path-unique set $M$ of $n$ points in the plane is segment-unique. [i]Cristi Săvescu[/i]

A kite with $AB = BC$ and $AD = CD$ has diagonals which satisfy $AC = 80$ and $BD = 71$. Let $AC$ and $BD$ intersect at a point $O$. Find the area of the quadrilateral formed by the circumcenters of $ABO$, $BCO$, $CDO$, and $ADO$.