Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
Let $a, b, c \geq 0$ and satisfy \[ a^2+b^2+c^2 +abc = 4 . \] Show that \[ 0 \le ab + bc + ca - abc \leq 2. \]
Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.
Let $S$ be a set of integers (not necessarily positive) such that
(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.
Prove that $S$ is the set of all integers.
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6
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).
I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that \[ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn \] for all $n > 0$.
Given that $x^2+5x+6=20$, find the value of $3x^2 + 15x+17$.
Express $\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}$ in the simplest possible form.
Let $r_1, \cdots, r_5$ be the roots of the polynomial $x^5 + 5x^4 - 79x^3 +64x^2 + 60x+144$. What is $r_1^2 +\cdots + r_5^2$?
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Find the greatest integer less than $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$$
Suppose that $P(x)$ is a polynomial with the property such that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is quintic (i.e. has a degree of $5$) with roots $r_1, \cdots, r_5$, find all the possible values of $|r_1|+|r_2|+\cdots+|r_5|$.
Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.
The Lucas numbers $L_n$ is defined as $L_0=2$, $L_1=1$, and $L_n=L_{n-1}+L_{n-2}$ for $n\ge 2$. Let $r=0.21347\dots$, whose digits are Lucas numbers. When numbers are multiple digits, they will "overlap", so $r=0.2134830\dots$, NOT $0.213471118\dots$. Express $r$ as a rational number $\frac{q}{p}$ where $p$ and $q$ are relatively prime.
The curve $y=x^4+2x^3-11x^2-13x+35$ has a bitangent (a line tangent to the curve at two points). What is the equation of this bitangent line.
Solve this equation $(x-2)(x+1)(x+4)(x+7)=19$.
Let real numbers $x, y,$ and $z$ satisfy $$x+\frac{1}{y}=4\quad\text{,}\quad y+\frac{1}{z}=1\quad\text{,}\quad z +\frac{1}{x}=\frac{7}{3}$$ Find the value of $xyz$.
Find the range of real number $a$ if equation $\mid\frac{x^2}{x-1}\mid=a$ has exactly two distinct real roots.
Solve this equation $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$