$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

The vertices of a $ 3 \minus{} 4 \minus{} 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? [asy]unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("3",midpoint(B--A),NW); label("4",midpoint(A--C),NE); label("5",midpoint(B--C),S);[/asy]$ \textbf{(A) } 12\pi\qquad \textbf{(B) } \frac {25\pi}{2}\qquad \textbf{(C) } 13\pi\qquad \textbf{(D) } \frac {27\pi}{2}\qquad \textbf{(E) } 14\pi$

Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21. A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$. [b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$. Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$ [b]p21.[/b] Let $N$ be the answer to Problem $20$. Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.

Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?

Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) [hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]'' I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url] [/hide]

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.

Find all values of the parameter $a$ for which the inequality \[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\] has a unique solution.

6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that \[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

Find all values of $a$ for which the inequality $$a^2x^2 + y^2 + z^2 \ge ayz+xy+xz$$ holds for all $x$, $y$ and $z$.

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$? [i]By Omid Hatami[/i]

Let $n$ be a positive integer. On the blackboard all quadratic polynomials with positive integer coefficients, that do not exceed $n$, without real roots are written Find all $n$ for which the number of written polynomials is even [i]A. Voidelevich[/i]

Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion \[(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.\]

Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$

The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and $$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$ Answer, with proof, to the following questions: a) What is the smallest possible value of $a$? b) What is the number of possible values of $a$?

Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.

For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + n + p + q$.

Given a finite set $X$, let $f$ be a rule such that $f$ maps every [i]even-element-subset[/i] $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions: (I) there exists an [i]even-element-subset[/i] $D$ of $X$ such that $f(D)>1990$; (II) for any two disjoint [i]even-element-subsets [/i]$A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds. Prove that there exist two subsets $P,Q$ of $X$ satisfying: (1) $P\cap Q=\emptyset$, $P\cup Q=X$; (2) for any [i]non-even-element-subset [/i]$S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$; (3) for any [i]even-element-subset[/i] $T$ of $Q$, we have $f(T)\le 1990$.