Find the minimum value of $$K(x,y)=16\frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$ where $x,y$ are the real allowed values

Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$ Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

[u]Round 1[/u] [b]p1.[/b] Suppose that gold satisfies the relation $p = v + v^2$, where $p$ is the price and $v$ is the volume. How many pieces of gold with volume $1$ can be bought for the price of a piece with volume $2$? [b]p2.[/b] Find the smallest prime number with each digit greater or equal to $8$. [b]p3.[/b] What fraction of regular hexagon $ZUMING$ is covered by both quadrilateral $ZUMI$ and quadrilateral$ MING$? [u]Round 2[/u] [b]p4.[/b] The two smallest positive integers expressible as the sum of two (not necessarily positive) perfect cubes are $1 = 1^3 +0^3$ and $2 = 1^3 +1^3$. Find the next smallest positive integer expressible in this form. [b]p5.[/b] In how many ways can the numbers $1, 2, 3,$ and $4$ be written in a row such that no two adjacent numbers differ by exactly $1$? [b]p6.[/b] A real number is placed in each cell of a grid with $3$ rows and $4$ columns. The average of the numbers in each column is $2016$, and the average of the numbers in each row is a constant $x$. Compute $x$. [u]Round 3[/u] [b]p7.[/b] Fardin is walking from his home to his oce at a speed of $1$ meter per second, expecting to arrive exactly on time. When he is halfway there, he realizes that he forgot to bring his pocketwatch, so he runs back to his house at $2$ meters per second. If he now decides to travel from his home to his office at $x$ meters per second, find the minimum $x$ that will allow him to be on time. [b]p8.[/b] In triangle $ABC$, the angle bisector of $\angle B$ intersects the perpendicular bisector of $AB$ at point $P$ on segment $AC$. Given that $\angle C = 60^o$, determine the measure of $\angle CPB$ in degrees. [b]p9.[/b] Katie colors each of the cells of a $6\times 6$ grid either black or white. From top to bottom, the number of black squares in each row are $1$, $2$, $3$, $4$, $5$, and $6$, respectively. From left to right, the number of black squares in each column are $6$, $5$, $4$, $3$, $2$, and $1$, respectively. In how many ways could Katie have colored the grid? [u]Round 4[/u] [b]p10.[/b] Lily stands at the origin of a number line. Each second, she either moves $2$ units to the right or $1$ unit to the left. At how many different places could she be after $2016$ seconds? [b]p11.[/b] There are $125$ politicians standing in a row. Each either always tells the truth or always lies. Furthermore, each politician (except the leftmost politician) claims that at least half of the people to his left always lie. Find the number of politicians that always lie. [b]p12.[/b] Two concentric circles with radii $2$ and $5$ are drawn on the plane. What is the side length of the largest square whose area is contained entirely by the region between the two circles? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934055p26256296]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

Fix integers $n\geq 2$ and $1\leq m\leq n-1$. Let $a_0, a_1, \dots, a_n$ be non-negative real numbers satisfying $a_0+a_1+\dots +a_n=1$. Prove that, if $\sum_{k=0}^n a_kx^k < x^m$ for some $0<x<1$, then $$\sum_{k=0}^{m-1}(m-k)a_k < \sum_{k=m+1}^n (k-m)a_k.$$

[b]p1.[/b] If Clark wants to divide $100$ pizzas among $25$ people so that each person receives the same number of pizzas, how many pizzas should each person receive? [b]p2.[/b] In a group of $3$ people, every pair of people shakes hands once. How many handshakes occur? [b]p3.[/b] Dylan and Joey have $14$ costumes in total. Dylan gives Joey $4$ costumes, and Joey now has the number of costumes that Dylan had before giving Joey any costumes. How many costumes does Dylan have now? [b]p4.[/b] At Banjo Borger, a burger costs $7$ dollars, a soda costs $2$ dollars, and a cookie costs $3$ dollars. Alex, Connor, and Tony each spent $11$ dollars on their order, but none of them got the same order. If Connor bought the most cookies, how many cookies did Connor buy? [b]p5.[/b] Joey, James, and Austin stand on a large, flat field. If the distance from Joey to James is $30$ and the distance from Austin to James is $18$, what is the minimal possible distance from Joey to Austin? [b]p6.[/b] If the first and third terms of a five-term arithmetic sequence are $3$ and $8$, respectively, what is the sum of all $5$ terms in the sequence? [b]p7.[/b] What is the area of the $S$-shaped figure below, which has constant vertical height $5$ and width $10$? [img]https://cdn.artofproblemsolving.com/attachments/3/c/5bbe638472c8ea8289b63d128cd6b449440244.png[/img] [b]p8.[/b] If the side length of square $A$ is $4$, what is the perimeter of square $B$, formed by connecting the midpoints of the sides of $A$? [b]p9.[/b] The Chan Shun Auditorium at UC Berkeley has room number $2050$. The number of seats in the auditorium is a factor of the room number, and there are between $150$ and $431$ seats, inclusive. What is the sum of all of the possible numbers of seats in Chan Shun Auditorium? [b]p10.[/b] Krishna has a positive integer $x$. He notices that $x^2$ has the same last digit as $x$. If Krishna knows that $x$ is a prime number less than $50$, how many possible values of $x$ are there? [b]p11.[/b] Jing Jing the Kangaroo starts on the number $1$. If she is at a positive integer $n$, she can either jump to $2n$ or to the sum of the digits of $n$. What is the smallest positive integer she cannot reach no matter how she jumps? [b]p12.[/b] Sylvia is $3$ units directly east of Druv and runs twice as fast as Druv. When a whistle blows, Druv runs directly north, and Sylvia runs along a straight line. If they meet at a point a distance $d$ units away from Druv's original location, what is the value of $d$? [b]p13.[/b] If $x$ is a real number such that $\sqrt{x} + \sqrt{10} = \sqrt{x + 20}$, compute $x$. [b]p14.[/b] Compute the number of rearrangments of the letters in $LATEX$ such that the letter $T$ comes before the letter $E$ and the letter $E$ comes before the letter $X$. For example, $TLEAX$ is a valid rearrangment, but $LAETX$ is not. [b]p15.[/b] How many integers $n$ greater than $2$ are there such that the degree measure of each interior angle of a regular $n$-gon is an even integer? [b]p16.[/b] Students are being assigned to faculty mentors in the Berkeley Math Department. If there are $7$ students and $3$ mentors and each student has exactly one mentor, in how many ways can students be assigned to mentors given that each mentor has at least one student? [b]p17.[/b] Karthik has a paper square of side length $2$. He folds the square along a crease that connects the midpoints of two opposite sides (as shown in the left diagram, where the dotted line indicates the fold). He takes the resulting rectangle and folds it such that one of its vertices lands on the vertex that is diagonally opposite. Find the area of Karthik's final figure. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01aa386f6616cafeed5f95ababb27bf24657f6.png[/img] [b]p18.[/b] Sally is inside a pen consisting of points $(a, b)$ such that $0 \le a, b \le 4$. If she is currently on the point $(x, y)$, she can move to either $(x, y + 1)$, $(x, y - 1)$, or $(x + 1, y)$. Given that she cannot revisit any point she has visited before, find the number of ways she can reach $(4, 4)$ from $(0, 0)$. [b]p19.[/b] An ant sits on the circumference of the circular base of a party hat (a cone without a circular base for the ant to walk on) of radius $2$ and height $\sqrt{5}$. If the ant wants to reach a point diametrically opposite of its current location on the hat, what is the minimum possible distance the ant needs to travel? [img]https://cdn.artofproblemsolving.com/attachments/3/4/6a7810b9862fd47106c3c275c96337ef6d23c2.png[/img] [b]p20.[/b] If $$f(x) = \frac{2^{19}x + 2^{20}}{ x^2 + 2^{20}x + 2^{20}}.$$ find the value of $f(1) + f(2) + f(4) + f(8) + ... + f(220)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$?

(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

Consider the following expression $$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$ Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).

Find the smallest positive integer $N$ such that each of the $101$ intervals $$[N^2, (N+1)^2), [(N+1)^2, (N+2)^2), \cdots, [(N+100)^2, (N+101)^2)$$ contains at least one multiple of $1001.$ [i]Proposed by Kyle Lee[/i]

Let $$A =\frac12 +\frac13 +\frac15 +\frac19,$$ $$B =\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{2 \cdot 9}+\frac{1}{3 \cdot 5}+\frac{1}{3 \cdot 9} +\frac{1}{5 \cdot 9},$$ $$C =\frac{1}{2 \cdot 3 \cdot 5} + \frac{1}{2 \cdot 3 \cdot 9} + \frac{1}{2 \cdot 5 \cdot 9} +\frac{1}{3 \cdot 5 \cdot 9}.$$ Compute the value of $A + B + C$.

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

Construct a set $S$ of polynomials inductively by the rules: (i) $x\in S$; (ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$. Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.

On two intersecting roads with equal constant speeds Audi and BMW cars are moving fast. It turned out that as in 17.00, and at 18.00 the BMW was twice as far from the intersection, than ''Audi''. At what time could an Audi drive across the river?