PRMO-2019

Problem 1 of 30

From a square with sides of length $5$, triangular pieces from the four corners are removed to form a regular octagon. Find the area $removed$ to the nearest integer.

Problem 2 of 30

Let $f(x)= x^2 + ax + b.$ If for all nonzero real $x$

$ f\left(x+\dfrac{1}{x}\right)=f(x)+f\left(\dfrac{1}{x}\right)$

and the roots of $f(x)=0$ are integers, what is the value of $a^2 +b^2\ ?$

Problem 3 of 30

Let $x_1$ be a positive real number and for every integer $N \ge 1$ let $x_{n+1} = 1 + x_1x_2...x_{n-1}x_n$. If $x_5 = 43$, what is the sum of digits of the largest prime factor of $x_6$?

Problem 4 of 30

An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. It then makes another $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this pattern until it reaches the anthill again, what is the distance in feet it would have walked?

Problem 5 of 30

Five persons wearing badges with numbers $1,\ 2,\ 3,\ 4,\ 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different.)

Problem 6 of 30

Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$?

Problem 7 of 30

On a clock, there are two instants between $12$ noon and $1$ PM, when the hour hand and the minute hand are at right angles.

The difference $in\ minutes$ between these two instants is written as $a + \dfrac{b}{c}$, where $a,\ b,\ c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a + b + c$?

Problem 8 of 30

How many positive integers $n$ are there such that $3 \le n \le 100$ and $x^{2^n} + x + 1$ is divisible by $x^2 + x + 1$?

Problem 9 of 30

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $\lt 100$. What is the value of $p - 3q$?

Problem 10 of 30

Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A$, $B$ and $C$ intersect $\Omega$ at $A_1$, $B_1$ and $C_1$ respectively, and the internal bisectors of angles $A_1$, $B_1$ and $C_1$ of the triangle $A_1B_1C_1$ intersect $\Omega$ at $A_2B_2C_2$, respectively. If the smallest angle of triangle $ABC$ is $40^\circ$, what is the magnitude of the smallest angle of $A_2B_2C_2$ in degrees?

Problem 11 of 30

How many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of angle $A,\ B$ and $C$ in degrees are positive integers and satisfy

$cos\ A\ cos\ B + sin\ A\ sin\ B\ sin\ kC = 1$

for some positive integer $k$, where $kC$ does not exceed $360^\circ$

Problem 12 of 30

A natural number $k \gt 1$ is called $good$ if there exist natural numbers

$a_1 \lt a_2 \lt \ ...\ \lt a_k$

such that

$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + ... + \dfrac{1}{\sqrt{a_k}} = 1$

Let $f(n)$ be the sum of the first $n$ $good\ numbers$, $n \ge 1$. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.

Problem 13 of 30

Each of the numbers $x_1,\ x_2,\ ...,\ x_{101}$ is $\pm 1$. What is the smallest positive value of

$\sum\limits_{1 \le i \lt j \le 101} x_ix_j$?

Problem 14 of 30

Find the smallest positive integer $n \ge 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

Problem 15 of 30

In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected?

Problem 16 of 30

A pen costs $Rs.\ 13$ and a notebook costs $Rs.\ 17$ A school spends exactly $Rs.\ 10000$ in the year $2017-18$ to buy $x$ pens and $y$ notebooks such that $x$ and $y$ are as close as possible (i.e. $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $Rs.\ 10000$ and buys $y$ pens and $x$ notebooks. How much $more$ did the school pay?

Problem 17 of 30

Find the number of ordered triples $(a,\ b,\ c)$ of positive integers such that $30a + 50b + 70c \le 343$.

Problem 18 of 30

How many ordered pairs of $(a,\ b)$ of positive integers with $a \lt b$ and $100 \le a,\ b \le 1000$ satisfy

$gcd(a,b):lcm(a,b) = 1:495$?

Problem 19 of 30

Let $AB$ be a diameter of a circle and let $C$ be a point on the segment $AB$ such that $AC:CB = 6:7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of triangle $XYZ$, find $[ABD]/[CDE]$ to the nearest integer.

Problem 20 of 30

Consider the set $E$ of natural numbers $n \le 10000$ such that when divided by $11,\ 12,\ 13$, respectively, the remainders, in that order, are distinct prime numbers in an arithmetic progression.

If $N$ is the largest number in $E$, find the sum of digits of $N$.

Problem 21 of 30

Consider the set $E = \{5,\ 6,\ 7,\ 8,\ 9\}$. For any partition of $\{A,\ B\}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number among these numbers. Find the sum of the digits of $N$.

Problem 22 of 30

What is the greatest integer not exceeding the sum $\sum\limits_{n = 1}^{1599} \dfrac{1}{\sqrt{n}}$?

Problem 23 of 30

Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of the $ABCD$ is the least.

If $\{PA,\ PB,\ PC,\ PD\} = \{3,\ 4,\ 6,\ 8\}$

what is the maximum possible area of $ABCD$?

Problem 24 of 30

A $1 \times n$ rectangle is divided into $n$ unit $(1 \times 1)$ squares. Each square of this rectangle is coloured red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of red squares can be zero).

Problem 25 of 30

A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village, $16\ m$ north of the north gate, and it can be just seen appearing on the horizon from a point $48\ m$ east of the south gate. What is the diameter, in meters, of the wall that surrounds the village?

Problem 26 of 30

Positive integers $x,\ y,\ z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.

Problem 27 of 30

We will say that a rearrangement of the letters of a word has $no\ fixed\ letters$ if, when the rearrangement is placed directly below the word, no column has the same letter repeated.

For instance $H\ B\ R\ A\ T\ A$ is a rearrangement with no fixed letters of $B\ H\ A\ R\ A\ T$ How many distinguishable rearrangements with no fixed letters does $B\ H\ A\ R\ A\ T$ have? (The two $A$s are considered identical.)

Problem 28 of 30

Let $ABC$ be a triangle with sides $51,\ 52,\ 53$. Let $\Omega$ denote the incircle of $\triangle ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1$, $r_2$, $r_3$ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed $r_1 + r_2 + r_3$.

Problem 29 of 30

In triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

Problem 30 of 30

Let $E$ denote the set of all natural numbers $n$ such that $3 \lt n \lt 100$ and the set $\{1,\ 2,\ 3,\ ...,\ n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.