Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$Here $d(n)$ is the number of positive divisors of a positive integer

Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented in infinitely many ways of the form $\pm P(1) \pm P(2) \pm \ldots \pm P(m)$, for some positive integer $m$ and certain choices of $\pm$.

Let $n > k$ be positive integers and let $F$ be a family of finite sets with the following properties: i. $F$ contains at least $\binom{n}{k}+ 1$ distinct sets containing exactly $k$ elements; ii. For any two sets $A, B \in F$ their union, i.e., $A \cup B$ also belongs to $F$. Prove that $F$ contains at least three sets with at least $n$ elements.

From a point $S$, which lies outside the circle $\Omega$, tangent lines $SA$ and $SB$ to that circle are drawn. On the chord $AB$ an arbitrary point $K$ is chosen. $SK$ intersects $\Omega$ at points $P$ and $Q$, and chords $RT$ and $UW$ pass through $K$ such that $W, Q$ and $T$ lie in the same half-plane with respect to $AB$. Lines $WR$ and $TU$ intersect chord $AB$ at $C$ and $D$, and $M$ is the midpoint of $PQ$. Prove that $\angle AMC = \angle BMD$

Republic has $n \geq 2$ cities, between some pairs of cities there are non-directed flight routes. From each city it is possible to get to any other city, and we will call the minimal number of flights required to do that the distance between the cities. For every city consider the biggest distance to another city. It turned out that for every city this number is equal to $m$. Find all values $m$ can attain for given $n$

Given two finite collections of pairs of real numbers It turned out that for any three pairs $(a_1, b_1)$, $(a_2, b_2)$ and $(a_3, b_3)$ from the first collection there is a pair $(c, d)$ from the second collection, such that the following three inequalities hold: \[ a_1c + b_1d \geq 0,a_2c + b_2c \geq 0 \text{ and } a_3c + b_3d \geq 0 \]Prove that there is a pair $(\gamma, \delta)$ in the second collection, such that for any pair $(\alpha, \beta)$ from the first collection inequality $\alpha \gamma + \beta \delta \geq 0$ holds.