Find all three-digit numbers that are \( 5 \) times greater than the product of their digits.

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60 ^\circ \qquad \textbf{(B)}\ 72 ^\circ\qquad \textbf{(C)}\ 75 ^\circ \qquad \textbf{(D)}\ 80 ^\circ\qquad \textbf{(E)}\ 90 ^\circ$

Prove that the four lines, joining the vertices of a tetrahedron with the incenters of the opposite faces, have a common point if and only if the three products of the lengths of opposite sides are equal.

2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$ ,:......., $A_n\setminus \{x\}$ are all distinct.

Suppose Robin and Eddy walk along a circular path with radius $r$ in the same direction. Robin makes a revolution around the circular path every $3$ minutes and Eddy makes a revolution every minute. Jack stands still at a distance $R>r$ from the center of the circular path. At time $t=0$, Robin and Eddy are at the same point on the path, and Jack, Robin, and Eddy, and the center of the path are collinear. When is the next time the three people (but not necessarily the center of the path) are collinear?

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

Find all real numbers $x$ such that the expression \[\log_2 |1 + \log_2 |2 + \log_2 |x| | |\] does not have a defined value.

Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .

A white plane is partitioned onto cells (in a usual way). A finite number of cells are coloured black. Each black cell has an even (0, 2 or 4) adjacent (by the side) white cells. Prove that one may colour each white cell in green or red such that every black cell will have equal number of red and green adjacent cells.

Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$. [color=#008000]Moderator says: and the original source for this one is Richard Stanley, [i]Enumerative Combinatorics[/i] vol. 1 (1st edition), exercise 1.15.[/color]

The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.

A man has a rectangular block of wood $m$ by $n$ by $r$ inches ($m, n,$ and $r$ are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do [i] not [/i] attempt to enumerate them.)

Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.

Let $\mathbb{L}$ be a finite field with $q$ elements. Prove that: a) If $q \equiv 3 \pmod 4$ and $n \ge 2$ is a positive integer divisible by $q-1,$ then $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times}.$ b) If there exists a positive integer $n \ge 2$ such that $x^n=(x^2+1)^n$ for all $x \in \mathbb{L}^{\times},$ then $q \equiv 3 \pmod 4$ and $q-1$ divides $n.$

Determine which number each letter denotes in the equalities $(YX)^Y=BYX$ and $(AA)^H = AHHA$, if different (identical) letters correspond to different (identical) numbers.

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^2/2$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

A number $k$ is [i]nice [/i] in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, . . . kn$ are each some cyclic shifts of the digits of $n$ in base $b$ (for example, $2$ is [i]nice [/i] in base $5$ because $2\cdot 135 = 315$). Determine all nice numbers in base $18$.

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

Let $H$ be the hyperbolic plane, $I(H)$ be the isometry group of $H$, and $O\in H$ be a fixed starting point. Determine those continuous $\sigma\colon H\rightarrow I(H)$ mappings that satisfty the following three conditions: (a) $\sigma(O)=\mathrm{id}$, and $\sigma (X)O=X$ for all $X\in H$; (b) for every $X\in H\backslash \{ O\}$ point, the $\sigma(X)$ isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant; (c) for any pair $P,Q\in H$ of points there exists a point $X\in H$ such that $\sigma(X)P=Q$. Prove that the $\sigma\colon H\rightarrow I(H)$ mappings satisfying the above conditions are differentiable with the exception of a point.