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 $k ?$
11
Check the two integers $k , k+1$ such that $2014$ lies between $k^2$ and ${(k+1)}^2$
Find the square root of $2014$ and find the value of $k$ , $k+1$
$k^2 < 2014 <{(k+1)}^2$ gives the value of $k$ as $44$.
Hence the prime factorization of $44$ gives $2\times 2\times 11$ . Hence the largest prime factor of $k$ is $11$
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?
Find the first term second term third term 4th term and check for any pattern
First term is 2014
Second term is ${2^3}+{0^3}+{1^3}+{4^3}=73$
Third term is ${7^3}+{3^3}=370$
Fourth term is ${3^3}+{7^3}=370$
We find that from third term onwards each term is $370$. So the $ {2014}^{th}$ is $370$
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 $D A ?$
Try to draw the diagram and apply Pythagoras theorem on each of the triangles.
In $\triangle AOB$ we have ${OA}^2+ {OB}^2=400......(i)$
In $\triangle BOC$ we have ${OB}^2+{OC}^2=4900.......(ii)$
In $\triangle COD$ we have ${OC}^2+{OD}^2=8100.....(iii)$
In $\triangle AOD$ we have ${OD}^2+{OA}^2={AD}^2.......(iv)$
From $(i) , (iii)$ we have ${OA}^2+{OB}^2+{OC}^2+{OD}^2=8500$
$({OA}^2+{OD}^2)+({OB}^2+{OC}^2=8500)$
${AD}^2+4900=8500$
${AD}^2=3600$ or $AD=60$
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?
Hint 1
The sum of two sides of a triangle is greater than the third side.
Assume the sides of a triangle to be of measurement $x$ and $3x$
Now $x+3x > 17 $ or $x > \frac{17}{4}$
$x+17> 3x$ give the equation $x<\frac{17}{2}$
$\frac{17}{4} < x <\frac{17}{2}$
Since $x$ is an integer hence $x = 5,6,7,8$
The maximum perimeter of the triangle is $8+24+17=49.$
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?
Find the product of the roots, and the sum of the roots.
Check the various combinations possible.
Find the least one among them.
Let the roots of the equation $x^2-nx+2014=0$ be $\alpha , \beta$
$\alpha \times \beta= 2014$
$\alpha+\beta=n$
Hence $2014 $ can be written as$(1\times 2014); (2\times 1007); (19\times 106); (38\times 53)$
The least possible value comes out to be $38+53= 91$
PRMO 2014, Problem 7:
If $x^{\left(x^{4}\right)}=4,$ what is the value of $x^{\left(x^{2}\right)}+x^{\left(x^{8}\right)} ?$
Hint 2
${x}^{x^4} = 4$ gives ${x}^4=1$ or ${x}^4=2^2$ or ${x}^4={\sqrt 2}^4$
Solution
Thus $x=\sqrt {2}, {x}^2=2, {x}^8=16$
Now ${x}^{x^{2}}+{x}^{x^{8}}$ gives ${\sqrt{2}}^2+{\sqrt{2}}^{16}=258$
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 $X Y ?$
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 $B \subseteq A ?$
