Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than $100$ pounds or more than $150$ pounds. So the boxes are weighed in pairs in every possible way. The results are $122$, $125$ and $127$ pounds. What is the combined weight in pounds of the three boxes? $\textbf{(A)}\ 160\qquad \textbf{(B)}\ 170\qquad \textbf{(C)}\ 187\qquad \textbf{(D)}\ 195\qquad \textbf{(E)}\ 354$

Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$

The second and fourth terms of a geometric sequence are $ 2$ and $ 6$. Which of the following is a possible first term? $ \textbf{(A)}\ \minus{}\!\sqrt3 \qquad \textbf{(B)}\ \minus{}\!\frac{2\sqrt3}{3} \qquad \textbf{(C)}\ \minus{}\!\frac{\sqrt3}{3} \qquad \textbf{(D)}\ \sqrt3 \qquad \textbf{(E)}\ 3$

Prove that $$ \sin^2 \alpha + \sin^2 \beta \geq \sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.$$

The edges of a cube are numbered from $1$ to $12$, then is calculated for each vertex the sum of the numbers of the edges going out from it. a) Prove that these sums cannot all be the same. b) Can eight equal sums result after one of the edge numbers is replaced by the number $13$ ?

Given is a triangle $ABC$. Let the points $P$ and $Q$ be on the sides $AB, AC$, respectively, so that $AP=AQ$, and $PQ$ passes through the incenter $I$. Let $(BPI)$ meet $(CQI)$ at $M$, $PM$ meets $BI$ at $D$ and $QM$ meets $CI$ at $E$. Prove that the line $MI$ passes through the midpoint of $DE$.

The Queen of Hearts rules a kingdom with $n$ (distinguishable) cities. Each pair of cities is either connected with a bridge or not connected with a bridge. Each day, the Queen of Hearts visits $2021$ cities. For every pair of cities, if she sees a bridge she gets angry and destroys it; otherwise she feels nice and constructs a bridge between them. We call two configurations of bridges [i]equivalent[/i] if one can be reached from the other after a finite number of days. Show that there is some integer $M$ such that if $n>M$, two configurations are equivalent if both of the following conditions hold: [list] [*] The parity of the total number of bridges is the same in both configurations [*] For every city, the parity of the number of bridges going out of that city is the same in both configurations. [/list]

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

Two different $3$ digit numbers are picked and then for every of them is calculated sum of all $5$ numbers which are getting when digits of picked number change place (etc. if one of the number is $707$, the sum is $2401=770+77+77+770+707$). Do the given results must be different?

a) A lamp of a lighthouse enlights an angle of $90$ degrees. Prove that you can turn the lamps of four arbitrary posed lighthouses so, that all the plane will be enlightened. b) There are eight lamps in eight points of the space. Each can enlighten an octant (three-faced space polygon with three mutually orthogonal edges). Prove that you can turn them so, that all the space will be enlightened.

We have placed some red and blue points along a circle. The following operations are permitted: (a) we may add a red point somewhere and switch the color of its neighbors, (b) we may take off a red point from somewhere and switch the color of its neighbors (if there are at least $3$ points on the circle and there is a red one too). Initially, there are two blue points on the circle. Using a number of these operations, can we reach a state with exactly two red point?

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 $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$?