PRMO-2016

Problem 1 of 16

Consider the sequence $1,\ 3,\ 3,\ 3,\ 5,\ 5,\ 5,\ 5,\ 5,\ 7,\ 7,\ 7,\ 7,\ 7,\ 7,\ 7,\ ...$ and evaluate its $2016\xasuper{th}$ term.

Problem 2 of 16

The five digit number $2a9b1$ is a perfect square. Find the value of $a^{b−1} + b^{a−1}$

Problem 3 of 16

The date index of a date is defined as $(12 \times month\ number + day\ number)$. 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.

Problem 4 of 16

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?

Problem 5 of 16

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?

Problem 6 of 16

Let $AD$ be an altitude in a right triangle $ABC$ with $\angle A = 90^\circ$ and $D$ on $BC$. Suppose that the radii of the incircles of the triangles $ABD$ and $ACD$ are $33$ and $56$ respectively. Let $r$ be the radius of the incircle of triangle $ABC$. Find the value of $3(r + 7)$.

Problem 7 of 16

Find the sum of digits in decimal form of the number $(999 . . . 9)^3$.

(There are $12$ nines)

Problem 8 of 16

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 $n = s(n) + p(n)$.

Problem 9 of 16

Suppose that $a$ and $b$ are real numbers such that $ab \ne 1$ and the equations $120a^2 − 120a + 1 = 0$ and $b^2 − 120b + 120 = 0$ hold. Find the value of $\dfrac{1+b+ab}{a}$.

Problem 10 of 16

Between $5$pm and $6$pm, 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.

Problem 11 of 16

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 $\dfrac{m}{n}$ where $m$, $n$ are relatively prime positive integers. Find the value of $m + n$.

Problem 12 of 16

Consider the 50 term sums:

$S = \dfrac{1}{1 \times 2} + \dfrac{1}{3 \times 4} + ... + \dfrac{1}{99 \times 100}$

$T = \dfrac{1}{51 \times 100} + \dfrac{1}{52 \times 99} + ... + \dfrac{1}{100 \times 51}$

The ratio $\dfrac{S}{T}$ is written in lowest form $\dfrac{m}{n}$ where $m$, $n$ are relatively prime natural numbers. Find the value of $m + n$.

Problem 13 of 16

Find the value of the expression:

$\dfrac{(3^4 + 3^2 + 1).(5^4 + 5^2 + 1).(7^4 + 7^2 + 1).(9^4 + 9^2 + 1)}{(2^4 + 2^2 + 1).(4^4 + 4^2 + 1).(6^4 + 6^2 + 1).(8^4 + 8^2 + 1)}$

$\ \ \ \times \dfrac{(11^4 + 11^2 + 1).(13^4 + 13^2 + 1)}{(10^4 + 10^2 + 1).(12^4 + 12^2 + 1)}$

when written in the lowest form.

Problem 14 of 16

The hexagon $OLYMPI$, with all sides equal, has a reflex angle at $O$ and convex at every other vertex. Suppose that $LP = 3\sqrt{2}$ units and the condition $\angle O = 10\angle L = 2\angle Y = 5\angle M = 2\angle P = 10\angle I$ holds. Find the area (in sq units) of the hexagon.

Problem 15 of 16

A natural number $a$ has four digits and $a^2$ ends with the same four digits as that of $a$. Find the value of $(10,080 − a)$.

Problem 16 of 16

Points $G$ and $O$ denote the centroid and the circumcenter of the triangle $ABC$. Suppose that $\angle AGO = 90^\circ$ and $AB = 17$, $AC = 19$. Find the value of $BC^2$.