INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 
Bose Olympiad Project Round is Live now. Learn More 

January 31, 2021

PRMO 2014 Problems & Solutions | Previous Year Paper

This post will provide you all the PRMO (Pre-Regional Mathematics Olympiad) 2014 problems and solutions. You may find some solutions with hints too.

PRMO 2014, Problem 1:

A natural number $k$ is such that $k^{2}<2014<(k+1)^{2}$. What is the largest prime factor of $k ?$

PRMO 2014, Problem 2:

The first term of a sequence is $2014 .$ Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014^{\text {th }}$ term of the sequence?

PRMO 2014, Problem 3:

Let $A B C D$ be a convex quadrilateral with perpendicular diagonals. If $A B=20, B C=70$ and $C D=90,$ then what is the value of $D A ?$

PRMO 2014, Problem 4:

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17 .$ What is the greatest possible perimeter of the triangle?

PRMO 2014, Problem 5:

If real numbers $a, b, c, d, e$ satisfy
what is the value of $a^{2}+b^{2}+c^{2}+d^{2}+e^{2} ?$

PRMO 2014, Problem 6:

What is the smallest possible natural number $n$ for which the equation $x^{2}-n x+2014=0$ has integer roots?

PRMO 2014, Problem 7:

If $x^{\left(x^{4}\right)}=4,$ what is the value of $x^{\left(x^{2}\right)}+x^{\left(x^{8}\right)} ?$

PRMO 2014, Problem 8:

Let. $S$ be a set of real numbers with mean $M$. If the means of the sets $S \cup{15}$ and $S \cup{15,1}$ are $M+2$ and $M+1$, respectively, then how many elements does $S$ have?

PRMO 2014, Problem 9:

Natural numbers $k, l, p$ and $q$ are such that if $a$ and $b$ are roots of $x^{2}-k x+l=0$ then $a+\frac{1}{b}$ and $b+\frac{1}{a}$ are the roots of $x^{2}-p x+q=0$. What is the sum of all possible values of $q$ ?

PRMO 2014, Problem 10:

In a triangle $A B C, X$ and $Y$ are points on the segments $A B$ and $A C,$ respectively, such that $A X: X B=1: 2$ and $A Y: Y C=2: 1$. If the area of triangle $A X Y$ is 10 then what is the area of triangle $A B \dot{C} ?$

PRMO 2014, Problem 11:

For natural numbers $x$ and $y,$ let $(x, y)$ denote the greatest common divisor of $x$ and $y .$ How many pairs of natural numbers $x$ and $y$ with $x \leq y$ satisfy the equation $x y=x+y+(x, y)$ ?

PRMO 2014, Problem 12:

Let $A B C D$ be a convex quadrilateral with $\angle D A B=\angle B D C=90^{\circ}$. Let the incircles of triangles $A B D$ and $B C D$ touch $B D$ at $P$ and $Q,$ respectively, with $P$ lying in between $B$ and $Q .$ If $A D=999$ and $P Q=200$ then what is the sum of the radii of the incircles of triangles $A B D$ and $B D C ?$

PRMO 2014, Problem 13:

For how many ratural numbers $n$ between 1 and 2014 (both inclusive) is $\frac{8 n}{9999-n}$ an integer?

PRMO 2014, Problem 14:

One morning, each member of Manjul's family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1 / 7$ -th of the total amount of milk and $2 / 17$ -th of the total amount of coffee. How many people are there in Manjul's family?

PRMO 2014, Problem 15:

Let $X O Y$ be a triangle with $\angle X O Y=90^{\circ} .$ Let $M$ and $N$ be the midpoints of legs $O X$ and OY, respectively. Suppose that $X N=19$ and $Y M=22 .$ What is $X Y ?$


PRMO 2014, Problem 16:

In a triangle $A B C,$ let $I$ denote the incenter. Let the lines $A I, B I$ and $C I$ intersect the incircle at $P, Q$ and $R$, respectively. If $\angle B A C=40^{\circ}$, what is the value of $\angle Q P R$ in degrees?

PRMO 2014, Problem 17:

For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^{2}+a x+b=0$ has integer roots. What is the smallest value of $b$ for which $N(b)=20 ?$

PRMO 2014, Problem 18:

Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(m n)=$ $f(m) f(n)$ for all natural numbers $m$ and $n .$ What is the least possible value of $f(999) ?$

PRMO 2014, Problem 19:

Let $x_{1}, x_{2}, \cdots, x_{2014}$ be real numbers different from $1,$ such that $x_{1}+x_{2}+\cdots+x_{2014}=1$ and
What is the value of
\frac{x_{1}^{2}}{1-x_{1}}+\frac{x_{2}^{2}}{1-x_{2}}+\frac{x_{3}^{2}}{1-x_{3}}+\cdots+\frac{x_{2014}^{2}}{1-x_{2014}} ?

PRMO 2014, Problem 20:

What is the number of ordered pairs $(A, B)$ where $A$ and $B$ are subsets of ${1,2, \ldots, 5}$ such that neither $A \subseteq B$ nor $B \subseteq A ?$

Some Useful Links:

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.