Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.

Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? $\textbf{(A)}\hspace{.05in} \dfrac1{24}\qquad \textbf{(B)}\hspace{.05in}\dfrac1{12} \qquad \textbf{(C)}\hspace{.05in}\dfrac18 \qquad \textbf{(D)}\hspace{.05in}\dfrac16 \qquad \textbf{(E)}\hspace{.05in}\dfrac14 $

Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z $). Of all the elegant triangles, which one has the smallest perimeter?

S-Corporation designs its logo by linking together $4$ semicircles along the diameter of a unit circle. Find the perimeter of the shaded portion of the logo. [img]https://cdn.artofproblemsolving.com/attachments/8/6/f0eabd46f5f3a5806d49012b2f871a453b9e7f.png[/img]

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

Let $a,$ $b,$ $c,$ $d$ be four positive reals such that $d=a+b+c+2\sqrt{ab+bc+ca}.$ Prove that $a=b+c+d-2\sqrt{bc+cd+db}.$ Darij Grinberg

Non-negative integers $a, b, c, d$ satisfy the equation $a + b + c + d = 100$ and there exists a non-negative integer n such that $$a+ n =b- n= c \times n = \frac{d}{n} $$ Find all 5-tuples $(a, b, c, d, n)$ satisfying all the conditions above.

A Metro ticket, which has six digits, is considered a "lucky number" if its six digits are different and their first three digits add up to the same as the last three (A number such as $026134$ is "lucky number"). Show that the sum of all the "lucky numbers" is divisible by $2002$.

The graph $G$ is not $3$-coloured. Prove that $G$ can be divided into two graphs $M$ and $N$ such that $M$ is not $2$-coloured and $N$ is not $1$-coloured. [i]V. Dolnikov[/i]

Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$. Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.

Prove that among any $10$ entries of the table $$0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, ... \,\,\,\, 9$$ $$9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, ... \,\,\,\, 8$$ $$8 \,\,\,\, 9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, ... \,\,\,\, 7$$ $$1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, 4 \,\,\,\, ... \,\,\,\, 0$$ standing in different rows and different columns, at least two are equal. (A Savin)

Consider the $ \mathbb {Z} / (10) $ additive group automorphism group of integers module $10$, that is, $ A = \left \{\phi: \mathbb {Z} / (10) \to \mathbb {Z} / (10) | \phi-automorphism \right \}$

Given $X \subset \mathbb{N}$, define $d(X)$ as the largest $c \in [0, 1]$ such that for any $a < c$ and $n_0\in \mathbb{N}$, there exists $m, r \in \mathbb{N}$ with $r \geq n_0$ and $\frac{\mid X \cap [m, m+r)\mid}{r} \geq a$. Let $E, F \subset \mathbb{N}$ such that $d(E)d(F) > 1/4$. Prove that for any prime $p$ and $k\in\mathbb{N}$, there exists $m \in E, n \in F$ such that $m\equiv n \pmod{p^k}$

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.

Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations \begin{align*}x(y^2+yz+z^2)&=3y+10z\\y(z^2+zx+x^2)&=21z+24x\\z(x^2+xy+y^2)&=7x+28y.\end{align*} Find $xy+yz+zx$.

Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that \[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\] and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set \[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\ 0 & \text{ otherwise}. \end{cases}\] \[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\ 0 & \text{ otherwise}. \end{cases}\] Prove that $ f_1$ and $ f_2$ are independent.

In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.

Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$