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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
$$a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3$$
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 $x y=x+y+(x, y)$ ?
Discussion

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 $X Y ?$

Discussion

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
$$
\frac{x_{1}}{1-x_{1}}+\frac{x_{2}}{1-x_{2}}+\cdots+\frac{x_{2014}}{1-x_{2014}}=1
$$
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 $B \subseteq A ?$

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