The lattice points $(i, j)$ for integers $0 \le i, j \le 3$ are each being painted orange or black. Suppose a coloring is good if for every set of integers $x_1, x_2, y_1, y_2$ such that $0 \le x_1 < x_2 \le 3$ and $0 \le y_1 < y_2 \le 3$, the points $(x_1, y_1),(x_1, y_2),(x_2, y_1),(x_2, y_2)$ are not all the same color. How many good colorings are possible?

In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point $D_1$ on the lines $A_1D$ and $C_1D$ respectively. a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$. b) Show that $\angle AQP = \angle D_1RS$.

Four perpendiculars are drawn from the vertices of a convex quadrangle to its diagonals. Prove that their bases make a quadrangle similar to the given one.

Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow: $x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$ $y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$ Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.

Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$. Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible? [i]Nikolai Beluhov[/i]

An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference $\frac56$: $$\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92.$$ The sequence of seven natural numbers $60$, $70$, $84$, $105$, $140$, $210$, $420$ has the property that the sequence inverted numbers (i.e. the row $\frac{1}{60}$, $\frac{1}{70}$, $\frac{1}{84}$, $\frac{1}{105}$, $\frac{1}{140}$, $\frac{1}{210}$,$\frac{1}{420}$ ) is an arithmetic sequence. (a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence? (b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?

Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

In the triangle $ABC$ the following equality holds: \[\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}\] Determine the value of $\frac{AC}{BC}$.

On the hypotenuse $AB$ of a right-angled triangle $ABC$, point $R$ is chosen, on the cathetus $BC$ a point $T$, and on the segment $AT$ a point $S$ are chosen in such a way that the angles $\angle ART$ and $\angle ASC$ are right angles. Points $M$ and $N$ are the midpoints of the segments $CB$ and $CR$, respectively. Prove that points $M$, $T$, $S$, and $N$ lie on the same circle. [i]Proposed by S. Berlov[/i]

In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: $07344$, $14098$, $27356$, $36429$, $45374$, $52207$, $63822$, $70558$, $85237$, $97665$. What is the combination to the safe?

Let $ f$ be a function from the non-negative integers to the non-negative integers such that $ f(nm) \equal{} n f(m) \plus{} m f(n), f(10) \equal{} 19, f(12) \equal{} 52,$ and $ f(15) \equal{} 26.$ What is $ f(8)$? A. 12 B. 24 C. 36 D. 48 E. 60

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]