Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?

[u]Set 6[/u] [b]p16.[/b] Let $n! = n \times (n - 1) \times ... \times 2 \times 1$. Find the maximum positive integer value of $x$ such that the quotient $\frac{160!}{160^x}$ is an integer. [b]p17.[/b] Let $\vartriangle OAB$ be a triangle with $\angle OAB = 90^o$ . Draw points $C, D, E, F, G$ in its plane so that $$\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,$$ and none of these triangles overlap. If points $O, A, G$ lie on the same line, then let $x$ be the sum of all possible values of $\frac{OG}{OA }$. Then, $x$ can be expressed in the form $m/n$ for relatively prime positive integers $m, n$. Compute $m + n$. [b]p18.[/b] Let $f(x)$ denote the least integer greater than or equal to $x^{\sqrt{x}}$. Compute $f(1)+f(2)+f(3)+f(4)$. [u]Set 7[/u] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \ge 0$. [b]p19.[/b] Find the least odd prime factor of $(F_3)^{20} + (F_4)^{20} + (F_5)^{20}$. [b]p20.[/b] Let $$S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...$$ Compute $420S$. [b]p21.[/b] Consider the number $$Q = 0.000101020305080130210340550890144... ,$$ the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to $\frac{1}{Q}$. [u]Set 8[/u] [b]p22.[/b] In five dimensional hyperspace, consider a hypercube $C_0$ of side length $2$. Around it, circumscribe a hypersphere $S_0$, so all $32$ vertices of $C_0$ are on the surface of $S_0$. Around $S_0$, circumscribe a hypercube $C_1$, so that $S_0$ is tangent to all hyperfaces of $C_1$. Continue in this same fashion for $S_1$, $C_2$, $S_2$, and so on. Find the side length of $C_4$. [b]p23.[/b] Suppose $\vartriangle ABC$ satisfies $AC = 10\sqrt2$, $BC = 15$, $\angle C = 45^o$. Let $D, E, F$ be the feet of the altitudes in $\vartriangle ABC$, and let $U, V , W$ be the points where the incircle of $\vartriangle DEF$ is tangent to the sides of $\vartriangle DEF$. Find the area of $\vartriangle UVW$. [b]p24.[/b] A polynomial $P(x)$ is called spicy if all of its coefficients are nonnegative integers less than $9$. How many spicy polynomials satisfy $P(3) = 2019$? [i]The next set will consist of three estimation problems.[/i] [u]Set 9[/u] Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed $1,000,000$. [b]p25.[/b] Suppose a circle of radius $20192019$ has area $A$. Let s be the side length of a square with area $A$. Compute the greatest integer less than or equal to $s$. If $n$ is the correct answer, an estimate of $e$ gives $\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}$ points. [b]p26.[/b] Given a $50 \times 50$ grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue? If $n$ is the correct answer, an estimate of $e$ gives $\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor$ points. [b]p27.[/b] The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is $\frac{\pi}{3\sqrt2} \approx 74.05\%$ of space (confirmed as recently as $2017!$), so we say that the packing density of spheres in three dimensions is about $0.74$. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than $10^8 \times d$. If $n$ is the correct answer, an estimate of e gives $\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}$ points. PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For certain $n$ countries there is an airline connecting any two countries, but some of the airlines are closed. Show that if the number of the closed airlines does not exceed $n-3$, then one can make a round trip using the remaining airlines, starting from one of the countries, visiting every country exactly once and returning to the starting country.

Quadrilateral $ABCD$ is cyclic and has positive integer side lengths. Suppose $AC \cdot BD = 53$ and $CD < DA$. The value of $\frac{AB /BC}{AD /DC}$ can be expressed as a common fraction $\frac{p}{q}$ , where $p$ and $q$ are relatively prime. Compute $p + q$.

Consider the numbers $x_1, x_2,...,x_n$ that satisfy: $\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$ $\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$ Prove that $n$ is a multiple of $4$.

Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$. (A. Blinkov)

The base $5$ number $32$ is equal to the base $7$ number $23$. There are two $3$-digit numbers in base $5$ which similarly have their digits reversed when expressed in base $7$. What is their sum, in base $5$? (You do not need to include the base $5$ subscript in your answer).

If $a,b,c$ are real positive numbers prove that the inequality holds: \[ \frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}} \]

let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$

Consider the figure below, composed of 16 segments. Prove that there is no curve intersecting each segment exactly once. (The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.) [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]

Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.

Find all positive integers $n$ satisfying the following. $$2^n-1 \text{ doesn't have a prime factor larger than } 7$$

A complex quartic polynomial $Q$ is [i]quirky [/i] if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x) = x^4-kx^3-x^2-x-45$ is quirky. Compute the product of these four values of $k$.

Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.

20) A particle of mass m moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. If the collision is completely $inelastic$ under what condition will the fractional momentum transfer between the two objects be a maximum? A) $\frac{m}{M} \ll 1$ B) $0.5 < \frac{m}{M} < 1$ C) $m = M$ D) $1 < \frac{m}{M} < 2$ E) $\frac{m}{M} \gg 1$

In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.

There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. (a) Prove that Geoff can always fulfill his wish if $n$ is odd. (b) Prove that Geoff can never fulfill his wish if $n$ is even.

Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by $$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$

Prove that the following statement holds for all odd integers $n \ge 3$: If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.