NMTC 2019 Stage II - Kaprekar (Class 7, 8) - Problems and Solutions

Problem 1

Let $a_n$ be the units place of $1^2+2^2+3^2+\ldots+n^2$. Prove that the decimal $0 . a_1 a_2 a_3 \ldots a_n \ldots$ is a rational number and represent it as $\frac{p}{q}$, where $p$ and $q$ are natural numbers.

Problem 2

(a) Find the positive integers $m, n$ such that $\frac{1}{m}+\frac{1}{n}=\frac{3}{17}$.
(b) Find the positive integers $m, n, p$ such that $\frac{1}{m}+\frac{1}{n}+\frac{1}{p}=\frac{3}{17}$.
(c) Using this idea, prove that we can find for any positive integer $k$, $k$ distinct integers, $n_1, n_2 \ldots . . n_k$ such that $\frac{1}{n_1}+\frac{1}{n_2}+\ldots \frac{1}{n_k}=\frac{3}{17}$.

Problem 3

Does there exist a positive integer which is a multiple of $2019$ and whose sum of the digits is $2019$ ? If no, prove it. If yes, give one such number.

Problem 4

In a triangle $XYZ$, the medians drawn through $X$ and $Y$ are perpendicular. Then show that $XY$ is the smallest side of $XYZ$.

Problem 5

Let $\triangle PQR$ be a triangle of area $1 cm^2$. Extend $QR$ to $X$ such that $QR=RX ; R P$ to $Y$ such that $R P=PY$ and $PQ$ to $Z$ such that $PQ=QZ$. Find the area of $\triangle XYZ$.

Problem 6

Find the real numbers $x$ and $y$ given that $x-y=\frac{3}{2}$ and $x^4+y^4=\frac{2657}{16}$.

Problem 7

The difference of the eight digit number $ABCDEFGH$ and the eight digit number $GHEFCDAB$ is divisible by $481$ . Prove that $C=E$ and $D=F$.

Problem 8

$ABCD$ is a parallelogram with area $36 \mathrm{~cm}^2$. $O$ is the intersection point of the diagonals of the parallelogram. $M$ is a point on $DC$. The intersection point of $AM$ and $BD$ is $E$ and the intersection point of $\mathrm{BM}$ and $\mathrm{AC}$ is $\mathrm{F}$. The sum of the areas of triangles $AED$ and $\mathrm{BFC}$ is $12 \mathrm{~cm}^2$. What is the area of the quadrilateral $EOFM$?