Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$. [asy] size(160); defaultpen(linewidth(0.6)); for(int j=0;j<=6;j=j+1) { draw((j,0)--(j,2)); } for(int i=0;i<=2;i=i+1) { draw((0,i)--(6,i)); } for(int k=1;k<=12;k=k+1) { label("$"+((string) k)+"$",(floor((k-1)/2)+0.5,k%2+0.5)); } [/asy]

Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then $$ tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.$$

You are bargaining with a salesperson for the price of an item. Your first offer is $a$ dollars and theirs is $b$ dollars. After you raise your offer by a certain percentage and they lower their offer by the same percentage, you arrive at an agreed price. What is that price, in terms of $a$ and $b$?

Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\] [i]Proposed by Karthik Vedula[/i]

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

Let $ x $ be a real number such that $ x^3 $ and $ x^2+x $ are rational numbers. Prove that $ x $ is rational.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.

[b]p1.[/b] Sherry starts with a three-digit positive integer. She subtracts $7$ from it, then multiplies the result by $7$, and then adds $7$ to that. If she ends up with $2023$, what number did she start with? [b]p2.[/b] Square $ABCD$ has side length $1$. Point $X$ lies on $\overline{AB}$ such that $\frac{AX}{XB} = 2$, and point $Y$ lies on $\overline{DX}$ such that $\frac{DY}{YX} = 3$. Compute the area of triangle $DAY$ . [b]p3.[/b] A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$? PS. You should use hide for answers.

Let $ABC$ be an equilateral triangle with $AB=a$ and $M\in(AB)$ a fixed point. Points $N\in(AC)$ and $P\in(BC)$ are taken such that the perimeter of $MNP$ is minimal. If the ratio between the areas of triangles $MNP$ and $ABC$ is $\textstyle\frac{7}{30},$ find the perimeter of triangle $MNP.$

Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$.

Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \]

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Compute \[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\] Here $i$ is the imaginary unit (that is, $i^2=-1$).

Find all the real solutions to $$n=\sum_{i=1}^n x_i=\sum_{1\le i<j\le n} x_ix_j$$ [i]Proposed by Carlos Dominguez, Valle[/i]

Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a [i]comparison[/i] will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ [i]comparisons[/i] are required.

A finite set of points $P$ in the plane has the following property: Every line through two points in $P$ contains at least one more point belonging to $P$. Prove that all points in $P$ lie on a straight line. [hide="Remark."]This may be a well known theorem called "Sylvester Gallai", but I didn't find this problem (I mean, exactly this one) using search function. So please discuss about the problem here, in this topic. Thanks :) [/hide]

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.

Let $ABC$ be an acute triangle. The altitudes $BE$ and $CF$ intersect at the orthocenter $H$, and point $O$ denotes the circumcenter. Point $P$ is chosen so that $\angle APH = \angle OPE = 90^{\circ}$, and point $Q$ is chosen so that $\angle AQH = \angle OQF = 90^{\circ}$. Lines $EP$ and $FQ$ meet at point $T$. Prove that points $A$, $T$, $O$ are collinear.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Evaluate $$ \sum_{k=0}^{n} \binom{n}{k}k^{2}.$$

Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.

Natural $a,b,c$ are $>100$ and $(a,b,c)=1$. $c|a+b,a|b+c$ Find minimal $b$