Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$

Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares. If we continue this process, the number of unit squares in the $ 100^\text{th}$ ring is $ \textbf{(A)}\ 396 \qquad \textbf{(B)}\ 404 \qquad \textbf{(C)}\ 800 \qquad \textbf{(D)}\ 10,\!000 \qquad \textbf{(E)}\ 10,\!404$ [asy]unitsize(3mm); defaultpen(linewidth(1pt)); fill((2,2)--(2,7)--(7,7)--(7,2)--cycle, mediumgray); fill((3,3)--(6,3)--(6,6)--(3,6)--cycle, gray); fill((4,4)--(5,4)--(5,5)--(4,5)--cycle, black); for(real i=0; i<=9; ++i) { draw((i,0)--(i,9)); draw((0,i)--(9,i)); }[/asy]

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy; $$P(n+d(n))=n+d(P(n)) $$ for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.

In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers?

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

Show that the sequence $\{a_n\}_{n\geq1}$ defined by $a_n = [n \sqrt 2]$ contains an infinite number of integer powers of $2$. ($[x]$ is the integer part of $x$.)

Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$.

Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Determine the real numbers $x$, $y$, $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

Points $H$ and $T$ are marked respectively on the sides $BC$ abd $AC$ of triangle $ABC$ so that $AH$ is the altitude and $BT$ is the bisectrix $ABC$. It is known that the gravity center of $ABC$ lies on the line $HT$. a) Find $AC$ if $BC$=a nad $AB$=c. b) Determine all possible values of $\frac{c}{b}$ for all triangles $ABC$ satisfying the given condition.

Find the number of ordered triples $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$?

For any positive integer $n$, consider its binary representation. Denote by $f(n)$ the number we get after removing all the $0$'s in its binary representation, and $g(n)$ the number of $1$'s in the binary representation. For example, $f(19) = 7$ and $g(19) = 3.$ Find all positive integers $n$ that satisfy $$n = f(n)^{g(n)}.$$ [i] Proposed by usjl[/i]

Let $\vartriangle ABC$ be a triangle. Let $D$ be the point on $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ be the point on the circumcircle of $ABC$ such that $DE$ is tangent to the circumcircle of $ABC$, but $E \ne A$. Let $F$ be the intersection of $AE$ and $BC$. Given that $BF/F C = 4/5$, find the maximum possible value for $\sin \angle ACB$/

In regular $n$-regular pyramid, the range value of dihedral angle of two adjacent sides is $\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)$

Let $A_1A_2A_3...A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set ${A_1A_2A_3...A_{12}}$

Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$, \[f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).\]

Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$. [i]Liubomir Chiriac, Moldova[/i]

When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one? $\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 47 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 49$

It is known that each rectangular parallelepiped has the following property: the square of its volume is equal to the product of areas of its three faces sharing a common vertex. Does there exist a parallelepiped which has the same property but is not rectangular? Alexandr Bufetov

Construct a triangle, by straight edge and compass, if the three points where the extensions of the medians intersect the circumcircle of the triangle are given.

(a) Find a polygon which can be cut by a straight line into two congruent parts so that one side of the polygon is divided in half while another side at a ratio of $1 : 2$. (b) Does there exist a convex polygon with this property?

Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.