PRMO (IOQM) 2020

Problem 1 of 30

Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be the midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?

(Here $[\Gamma]$ denotes the area of the geometrical figure $\Gamma$.)

Problem 2 of 30

A number $N$ in base $10$, is $503$ in base $b$ and $305$ in base $b+2$. What is the product of the digits of $N$?

Problem 3 of 30

If $\sum\limits_{k=1}^{N}\dfrac{2k + 1}{(k^2 + k)^2} = 0.9999$ then determine the value of $N$

Problem 4 of 30

Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the mid-point of the side $BC$. Find the area of the rectangle.

Problem 5 of 30

Find the number of integer solutions to ${\Large{|}}|x| - 2020{\Large{|}} \lt 5$.

Problem 6 of 30

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square.

Problem 7 of 30

Let $ABC$ be a triangle with $AB = AC$. Let $D$ be a point on the segment $BC$ such that $BD = 48\dfrac{1}{61}$ and $DC = 61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE = 11$. Find $AE$.

Problem 8 of 30

A $5$-digit number (in base $10$) has digits $k$, $k + 1$, $k + 2$, $3k$, $k + 3$ in that order, from left to right. If this number if $m^2$ for some natural number $m$, find the sum of the digits of $m$.

Problem 9 of 30

Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, $BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM \parallel AC$ and $DN \parallel BC$. If $(MN)^2 = \dfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?

Problem 10 of 30

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? (The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are and even number of scores.)

Problem 11 of 30

Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and

$S = \{(a,b) \in X \times X : x^2 + ax + b$ and $x^3 + bx + a$ have at least a common real zero$\}$.

How many elements are there in $S$?

Problem 12 of 30

Given a pair of concentric circles, chords $AB$, $BC$, $CD,\ ...$ of the outer circle are drawn such that they all touch the inner circle.

If $\angle ABC = 75^\circ$, how many chords can be drawn before returning to the starting point?

Problem 13 of 30

Find the sum of all positive integers $n$ for which $|2^n + 5^n - 65|$ is a perfect square.

Problem 14 of 30

The product $55 \times 60 \times 65$ is written as the product of five distinct positive integers. What is the least possible value of the largest of these integers?

Problem 15 of 30

Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many arrangements are possible?

Problem 16 of 30

The sides $x$ and $y$ of a scalene triangle satisfy $x + \dfrac{2\Delta}{x} = y + \dfrac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. If $x = 60$, $y = 63$, what is the length of the largest side of the triangle?

Problem 17 of 30

How many two digit numbers have exactly $4$ positive factors? (Here $1$ and the number $n$ itself are also considered as factors of $n$.)

Problem 18 of 30

If,

$\sum\limits_{k = 1}^{40}\left(\sqrt{1 + \dfrac{1}{k^2} + \dfrac{1}{(k+1)^2}}\right) = a + \dfrac{b}{c}$

where $a, b, c \in \mathbb{N}, b \lt c, gcd(b, c) = 1$, then what is the value of $a + b$

Problem 19 of 30

Let $ABCD$ be a parallelogram. Let $E$ and $F$ be midpoints of $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect in $P$ and form four triangles $APB$, $BPC$ and $CPD$ and $DPA$. If the area of the parallelogram is $100$ sq. units, what is the maximum area in sq. units of a triangle among this four triangles?

Problem 20 of 30

A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked $5$ times as many hours as the last woman, for how many hours did the first woman work?

Problem 21 of 30

A total fixed amount of $N$ thousand rupees is given to three persons $A$, $B$, $C$, every year, each being given an amount proportional to her age. In the first year, $A$ got half the total amount. When the sixth payment was made, $A$ got six-seventh of the amount that she had in the first year; $B$ got $Rs\ 1000$ less than that she had in the first year; and $C$ got twice of that she had in the first year. Find $N$.

Problem 22 of 30

In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7$, $QR = 6$ and $RS = 8$ what is the area of triangle $ABC$?

Problem 23 of 30

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

Problem 24 of 30

A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (a circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the $X$ axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m$, $n$ are positive integers and $n$ is square-free. Find $m + n$.

Problem 25 of 30

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that

$\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N$

find the sum of the squares of the digits of $N$.

Problem 26 of 30

In the figure below, $4$ of the $6$ disks are to be colored black and $2$ are to be colored white. Two colorings that can be obtained by a rotation or a reflection of the entire figure are considered the same.

There are only four such colorings for the given two colors, as shown in Figure 1.

In how many ways can we color the $6$ disks such that $2$ are colored black, 2 are colored white, $2$ are colored blue with the given identification condition?

Problem 27 of 30

A bug travels in the coordinate plane moving only along the lines that are parallel to the $x$ axis or $y$ axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length atmost $14$. How many points with integer coordinates lie on at least one of these paths.

Problem 28 of 30

A natural number $n$ is said to be good if $n$ is the sum of $r$ consecutive positive integers, for some $r \ge 2$. Find the number of good numbers in the set $\{1, 2, ... , 100\}$.

Problem 29 of 30

Positive integers $a, b, c$ satisfy $\dfrac{ab}{a - b} = c$. What is the largest possible value of $a + b + c$ not exceeding $99$?

Problem 30 of 30

Find the number of pairs $(a, b)$ of natural numbers such that $b$ is a $3$-digit number, $a+1$ divides $b - 1$ and $b$ divides $a^2 + a + 2$.