A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.

Above the segments $AB$ and $BC$ we drew a semicircle at each. $F_1$ bisects $AB$ and $F_2$ bisects $BC$. Above the segments $AF_2$ and $F_1C$ we also drew a semicircle at each. Segments $P Q$ and $RS$ touch the corresponding semicircles as shown in the figure. Prove that $P Q \parallel RS$ and $|P Q| = 2 \cdot |RS|$. [img]https://cdn.artofproblemsolving.com/attachments/8/2/570e923b91e9e630e3880a014cc6df4dc33aa2.png[/img]

The number $ 25^{64}\cdot64^{25}$ is the square of a positive integer $ N$. In decimal representation, the sum of the digits of $ N$ is $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 35$

In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$. Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$. Prove that: \[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \] Darij

Let $[ {x} ]$ be the greatest integer less than or equal to $x$, and let $\{x\}=x-[x]$. Solve the equation: $[x] \cdot \{x\} = 2005x$

$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\] is a rational number. Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows: $$g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\ 0 &\text{if} ~ x = 0. \end{cases}$$ Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]

A point$ O$ inside a parallelogram $ABCD$ satisfies $\angle AOB + \angle COD = \pi$. Prove that $\angle CBO = \angle CDO$.

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.

A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.

Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.

The points $ A,B,K,L,X$ lies of the circle $\Gamma$ in that order such that the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. The circle that passes through $A$ and tangent to $BK$ at $B$ intersects the line segment $KX$ at $P$ and $Q$. The circle that passes through $A$ and tangent to $BL$ at $B$ intersect the line segment $BX$ for the second time at $T$. Prove that $\angle{PTB} = \angle{XTQ}$

Suppose sequence $\{a_i\} = a_1, a_2, a_3, ....$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot ...\cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$ .

Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$. Prove that the lines $BK$ and $AC$ are perpendicular. [i](M. Karpuk)[/i]

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table. Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 min(a,b,c)$. After $672$ times, on the table there is only one number $m$, such that (A): $0 < m < 1$ (B): $m = 1$ (C): $1 < m < 2$ (D): $m = 2$ (E): None of the above.

From time $t = 0$ to time $t = 1$ a population increased by $i\%$, and from time $t = 1$ to time $t = 2$ the population increased by $j\%$. Therefore, from time $t = 0$ to time $t = 2$ the population increased by $ \textbf{(A)}\ (i + j)\%\qquad\textbf{(B)}\ ij\%\qquad\textbf{(C)}\ (i+ij)\%\qquad\textbf{(D)}\ \left(i + j + \frac{ij}{100}\right)\%\qquad\textbf{(E)}\left( i + j + \frac{i + j}{100}\right)\% $

Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

There are 10 stones of different weights with distinct pairwise sums. We have a special two-tiered balance scale such that only two stones can be put on each cup and then we understand which cup is heavier. Prove that having this scale you can either find the heaviest or the lightest stone.

The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.

Let $ABC$ be an isosceles right triangle with hypotenuse $AB$, $D$ be some point in the plane such that $2CD = AB$ and point $C$ inside the triangle $ABD$. We construct two rays with a start in $C$, intersecting $AD$ and $BD$ and perpendicular to them. On the first one, intersecting $AD$, we will plot the segment $CK = AD$, and on the second one - $CM = BD$. Prove that points $M$, $D$ and $K$ lie on the same line.