MOCK TEST UGA

Let $A=\{\frac{a}{b}\mid a|1000 \mbox{ and } b|1000 \mbox{ and } (a,b)=1, a,b \in \mathbb{Z}^+\}.$ Let the sum of all the elements in $A$ be denoted by $Sum$. What is $[\frac{Sum}{10}]$? (where$ [x]$ denotes the greatest integer less than or equal to $x$).

Let $a$ and $b$ be positive integers less than $10^6$ and they satisfy the equation below
$$
a+b=2\sqrt{ab}+4
$$
Find the number of ordered pairs $(a,b)$.

If the coefficients of $x^{2}$ and $x^{3}$ are equal when we expand $(px + q)^{2000},$ where $p, q \in \mathbb{Z}^+$ and $(p,q)=1$. Find $p + q$.

Find the sum of the digits of the least positive integer $n$ such that, if we write $10^n=a \times b$ where $a,b\in \mathbb{Z}^+$, then for any such factorization of $10^n$ either $a$ or $b$ will contain the digit $0$.

The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to 10 . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m, n$ are relatively prime integers. Find $m+n$.

Given that
$$
\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}
$$
find the greatest integer that is less than $\frac N{100}$.

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find $n$.

In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001}$, find $BC^2$.

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

In Lotto, 6 numbers are chosen from $\{1,2, \dots, 49\}$ with $\binom{49}{6}$ possible $6$-subsets. How many of these subsets have at least a pair of neighbours?

What is the minimum number of pairwise comparisons needed to identify the heaviest and the second heaviest of 128 objects?

Find the sum of
$$
1+\frac{1}{1+2}+\frac{1}{1+2+3}+\dots+\frac{1}{1+2+3+\dots+n}
$$

Given that
$$
(1 + \tan 1^{\circ})(1 + \tan 2^\circ)\dots(1 + \tan {45}^\circ) = 2^n,
$$
find $n.$

Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p)=h(x)(x-p)^{2}$ for all $x \in \mathbb{R}$. Now, consider the polynomial
$$
f(x)=12 x^{5}-15 x^{4}-40 x^{3}+540 x^{2}-2160 x+1
$$
and suppose that it’s double attractors are $a_{1}, a_{2}, \ldots, a_{n}$. If the sum $\sum_{i=1}^{n}\left|a_{i}\right|$ can be written as $\sqrt{a}+\sqrt{b}$, where $a, b$ are positive integers, find $a+b$.

Find the least positive integer $n$ such that

$$
\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.
$$

The sum $\sum_{j=1}^{2021}\left|\sin \frac{2 \pi j}{2021}\right|$ can be written as $\tan \left(\frac{c \pi}{d}\right)$ for some relatively prime positive integers $c, d$, such that $2 * c<d$. Find the value of $c+d$.

If the equation
$$
\sum_{i=1}^n (x+i-1)(x+i)=10n
$$
has two roots which are consecutive integers then find $n$.

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$, on which the normals are parallel to the line $2x-y=1$

Find the value of $\lambda$, if the lines $3x-4y-13=0,8x-11y-33$, and $2x-3y+\lambda =0$ are concurrent.

If $f(n)= \lim_{x \to 0}\bigg\{\bigg(1+\sin\frac{x}{2}\bigg)\bigg(1+\sin\frac{x}{2^2}\bigg)\dots \bigg(1+\sin\frac{x}{2^n}\bigg)\bigg\}^\frac{1}{x}$\\
then find $\lim_{n\to\infty} f(n)$.

Evaluate $\lim_{x\to 0}\frac{e^{\sin x}-(1+\sin x)}{(\tan(\sin x))^2}$.

Let $f(x)=1+2 x+3 x^{2}+4 x^{3}+5 x^{4}$ and let $\zeta=e^{2 \pi i / 5}=\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}$. Find the value of the following expression:
$$
f(\zeta) f\left(\zeta^{2}\right) f\left(\zeta^{3}\right) f\left(\zeta^{4}\right)
$$

Determine the greatest integer less than or equal to
$$
100 \sum_{n=0}^{\infty} \frac{1}{(n+3) \cdot n !} .
$$

Given
$$
4 \int_{\ln 3}^{\ln 5} \frac{e^{3 x}}{e^{2 x}-2 e^{x}+1} d x=a+b \ln 2
$$
where $a$ and $b$ are integers, what is the value of $a+b$ ?

The double factorial of a positive integer $n$ is denoted $n ! !$ and equals the product $n(n-2)(n-$ 4) $\cdots\left(n-2\left(\left\lceil\frac{n}{2}\right\rceil-1\right)\right)$; we further specify that $0 ! !=1$. What is the greatest integer $q$ such that
$$
\sqrt[4]{q}<\sum_{n=0}^{\infty} \frac{1}{(2 n) ! !} ?
$$

Let $A=\int^{\infty}_0\frac{\log x}{1+x^3}dx$. Then find the value of $\int^{\infty}_0\frac{x\log x}{1+x^3}dx$ in terms of $A$.

A running track of $440$ ft is to be laid out enclosing a foootball field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then find the lengths of its sides.

Let $e$ be Euler’s constant. For all real $x$ greater than $e$, let $f(x)$ be the unique positive real value $y$ satisfying $y<x$ and $x^{y}=y^{x}$. Over $x \in(e, \infty)$, the function $y=f(x)$ is differentiable, and the value of $f^{\prime}(4)$ can be expressed as $\frac{1}{a}-\frac{1}{b-\ln c}$ for positive integers $a, b$, and $c$. Compute the value of $a+b+c$.

For a bijective function $g: \mathbb{R} \rightarrow \mathbb{R}$, we say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is its superinverse if it satisfies the following identity $(f \circ g)(x)=g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x)=x^{3}+9 x^{2}+27 x+81$ and $f$ is its superinverse, find $|f(-289)|$.