PRMO - 2016 - Questions, Discussions, Hints, Solutions

This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you.

1.	2.	3.	4.	5.
6.	7.	8.	9.	10.
11.	12.	13.	14.	15.
16.	17.	18.	19.	20.
21.	22.	23.	24.	25.
26.	27.	28.	29.	30.

Problems

Consider the sequence 1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, ... and evaluate its 2016th term
Sequence and series
The five digit number $2a9b1$ is a perfect square. Find the value of $a^{b−1} + b^{a−1}$ .
Number Theory
The date index of a date is defined as (12 × month number + day number). Three events each with a frequency of once in 21 days, 32 days and 9 days, respectively, occurred simultaneously for the first time on July 31, 1961 (Ireland joining the European Economic Community). Find the date index of the date when they occur simultaneously for the eleventh time.
Number Theory
There are three kinds of fruits in the market. How many ways are there to purchase 25 fruits from among them if each kind has at least 25 of its fruit available?
Combinatorics
In a school there are 500 students. Two-thirds of the students who do not wear glasses, do not bring lunch. Three-quarters of the students who do not bring lunch do not wear glasses. Altogether, 60 students who wear glasses bring lunch. How many students do not wear glasses and do not bring lunch?
Set Theory
Let AD be an altitude in a right triangle ABC with ∠A = 90◦ and D on BC. Suppose that the radii of the in-circles of the triangles ABD and ACD are 33 and 56 respectively. Let r be the radius of the in-circle of triangle ABC. Find the value of $3(r + 7)$ .
Geometry
or,
Find the sum of digits in decimal form of the number $(999 . . . 9)^3$ . (There are 12 nines).
Let $s(n)$ and $p(n)$ denote the sum of all digits of n and the product of all digits of n (when written in decimal form), respectively. Find the sum of all two-digit natural numbers n such that $n = s(n) + p(n)$ .
Number Theory
Suppose that a and b are real numbers such that $ab\not= 1$ and the equations $120 a^2 − 120 a + 1 = 0$ and $b^2 − 120b + 120 = 0$ hold. Find the value of $\frac{1+b+ab}{a}$ .
Quadratic equation
Between 5pm and 6pm, I looked at my watch. Mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch.
Number theory
In triangle ABC right angled at vertex B, a point O is chosen on the side BC such that the circle $\gamma$ centered at O of radius OB touches the side AC. Let AB = 63 and BC = 16, and the radius of $\gamma$ be of the form $\frac{m}{n}$ where m, n are relatively prime positive integers. Find the value of $m+n$ .
Geometry
Consider the 50 term sums:
$S=\frac{1}{1*2}+ \frac{1}{3*4} +......+ \frac{1}{99*100}$
$T= \frac{1}{51*100}+ \frac{1}{52*99}+..........+ \frac{1}{100*51}$
The ratio $S/T$ is written in the lowest form $m/n$ where m,n are relatively prime natural numbers. Find the value of $m+n$ .
Sequence and Series
Find the value of the expression
$\frac{\left(3^{4}+3^{2}+1\right) \cdot\left(5^{4}+5^{2}+1\right) \cdot\left(7^{4}+7^{2}+1\right) \cdot\left(9^{4}+9^{2}+1\right) \cdot\left(11^{4}+11^{2}+1\right) \cdot\left(13^{4}+13^{2}+1\right)}{\left(2^{4}+2^{2}+1\right) \cdot\left(4^{4}+4^{2}+1\right) \cdot\left(6^{4}+6^{2}+1\right) \cdot\left(8^{4}+8^{2}+1\right) \cdot\left(10^{4}+10^{2}+1\right) \cdot\left(12^{4}+12^{2}+1\right)}$
When Written in the lowest form.
Arithmetic Calculation
The hexagon $OLYMPI$ has a reflex angle at O and convex at every other vertex. Suppose that $LP = 3\sqrt 2$ units and the condition $\angle O = 10\angle L = 2\angle Y = 5∠M = 2 \angle P = 10 \angle I$ holds. Find the area (in sq units) of the hexagon.
Geometry
A natural number a has four digits and $a^2$ ends with the same four digits as that of a. Find the value of $(10,080 − a)$ .
Number Theory
Points G and O denote the centroid and the circumcenter of the triangle ABC. Suppose that ∠AGO = 90◦ and AB = 17,AC = 19. Find the value of $BC^2$ .
Geometry

THE END

This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you.

1.	2.	3.	4.	5.
6.	7.	8.	9.	10.
11.	12.	13.	14.	15.
16.	17.	18.	19.	20.
21.	22.	23.	24.	25.
26.	27.	28.	29.	30.

Problems

Consider the sequence 1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, ... and evaluate its 2016th term
Sequence and series
The five digit number $2a9b1$ is a perfect square. Find the value of $a^{b−1} + b^{a−1}$ .
Number Theory
The date index of a date is defined as (12 × month number + day number). Three events each with a frequency of once in 21 days, 32 days and 9 days, respectively, occurred simultaneously for the first time on July 31, 1961 (Ireland joining the European Economic Community). Find the date index of the date when they occur simultaneously for the eleventh time.
Number Theory
There are three kinds of fruits in the market. How many ways are there to purchase 25 fruits from among them if each kind has at least 25 of its fruit available?
Combinatorics
In a school there are 500 students. Two-thirds of the students who do not wear glasses, do not bring lunch. Three-quarters of the students who do not bring lunch do not wear glasses. Altogether, 60 students who wear glasses bring lunch. How many students do not wear glasses and do not bring lunch?
Set Theory
Let AD be an altitude in a right triangle ABC with ∠A = 90◦ and D on BC. Suppose that the radii of the in-circles of the triangles ABD and ACD are 33 and 56 respectively. Let r be the radius of the in-circle of triangle ABC. Find the value of $3(r + 7)$ .
Geometry
or,
Find the sum of digits in decimal form of the number $(999 . . . 9)^3$ . (There are 12 nines).
Let $s(n)$ and $p(n)$ denote the sum of all digits of n and the product of all digits of n (when written in decimal form), respectively. Find the sum of all two-digit natural numbers n such that $n = s(n) + p(n)$ .
Number Theory
Suppose that a and b are real numbers such that $ab\not= 1$ and the equations $120 a^2 − 120 a + 1 = 0$ and $b^2 − 120b + 120 = 0$ hold. Find the value of $\frac{1+b+ab}{a}$ .
Quadratic equation
Between 5pm and 6pm, I looked at my watch. Mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch.
Number theory
In triangle ABC right angled at vertex B, a point O is chosen on the side BC such that the circle $\gamma$ centered at O of radius OB touches the side AC. Let AB = 63 and BC = 16, and the radius of $\gamma$ be of the form $\frac{m}{n}$ where m, n are relatively prime positive integers. Find the value of $m+n$ .
Geometry
Consider the 50 term sums:
$S=\frac{1}{1*2}+ \frac{1}{3*4} +......+ \frac{1}{99*100}$
$T= \frac{1}{51*100}+ \frac{1}{52*99}+..........+ \frac{1}{100*51}$
The ratio $S/T$ is written in the lowest form $m/n$ where m,n are relatively prime natural numbers. Find the value of $m+n$ .
Sequence and Series
Find the value of the expression
$\frac{\left(3^{4}+3^{2}+1\right) \cdot\left(5^{4}+5^{2}+1\right) \cdot\left(7^{4}+7^{2}+1\right) \cdot\left(9^{4}+9^{2}+1\right) \cdot\left(11^{4}+11^{2}+1\right) \cdot\left(13^{4}+13^{2}+1\right)}{\left(2^{4}+2^{2}+1\right) \cdot\left(4^{4}+4^{2}+1\right) \cdot\left(6^{4}+6^{2}+1\right) \cdot\left(8^{4}+8^{2}+1\right) \cdot\left(10^{4}+10^{2}+1\right) \cdot\left(12^{4}+12^{2}+1\right)}$
When Written in the lowest form.
Arithmetic Calculation
The hexagon $OLYMPI$ has a reflex angle at O and convex at every other vertex. Suppose that $LP = 3\sqrt 2$ units and the condition $\angle O = 10\angle L = 2\angle Y = 5∠M = 2 \angle P = 10 \angle I$ holds. Find the area (in sq units) of the hexagon.
Geometry
A natural number a has four digits and $a^2$ ends with the same four digits as that of a. Find the value of $(10,080 − a)$ .
Number Theory
Points G and O denote the centroid and the circumcenter of the triangle ABC. Suppose that ∠AGO = 90◦ and AB = 17,AC = 19. Find the value of $BC^2$ .
Geometry

THE END