Mock Test UGB

Suppose you are given with the set, $S = \{1, 2, 3, …, 2005\}$. Find the minimum $n$ such that whenever you select any $n$ co-prime numbers from the set $S$ then you will always have a prime number in your collection. (Please note that $1$ is not considered to be a prime number).

For all positive integers $n$, let

$$
f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.
$$

Compute $f(2019) – f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

Define functions $f_0, f_1, \dots,f_n, \dots$ to satisfy the following relations: $f_0=2, \frac{d}{dx}f_n=f_{n-1}.$ If for $n \geq 1, f_n(0)=0,$ compute

$$
\sum_{n=0}^{\infty} f_n(ln(2019)).
$$

Let $A B C$ be an acute angled triangle with circumcircle $\Gamma$. Let $l_{B}$ and $l_{C}$ be the lines perpendicular to $B C$ which pass through $B$ and $C$ respectively. A point $T$ lies on the minor arc $B C$. The tangent to $\Gamma$ at $T$ meets $l_{B}$ and $l_{C}$ at $P_{B}$ and $P_{C}$ respectively. The line through $P_{B}$ perpendicular to $A C$ and the line through $P_{C}$ perpendicular to $A B$ meet at a point $Q$. Given that $Q$ lies on $B C$, prove that the line $A T$ passes through $Q$.
(A minor arc of a circle is the shorter of the two arcs with given endpoints.)

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.

Assume that $a_0,a_1,\dots,a_n$ are real number such that

$$
\frac{a_0}{n+1}+\frac{a_1}{n}+\dots+ \frac{a_{n-1}}{2}+a_n=0.
$$

Prove that the polynomial $P(x)=a_0x^n+a_1x^{n-1}+\dots+a_n$ has at least one root in $(0,1)$.

In how many ways can you select two disjoint subsets from an $n$-set?Consider only the unordered pairs of subsets.

A function $f:(a,b)\to\mathbb{R}$ is continuous. Prove that, given $x_1,x_2,\dots, x_n$ is $(a,b)$, there exists $x_0\in(a,b)$ such that

$$
f(x_0)=\frac{1}{n}(f(x_1)+f(x_2)+\dots+ f(x_n)).
$$