NMTC Stage 2 KAPREKAR Sub Junior Level 2026

A rectangular box is rested on a horizontal surface. Length, breadth and height of the box are $$20\text{ cm}$$, $$12\text{ cm}$$ and $$9\text{ cm}$$ respectively. The box has 5 faces; it is open from the top. Its vertices are named $$A, B, C, D, P, Q, R, S$$ as shown in the figure. There are two ants; ant-1 at the vertex $$A$$ and ant-2 at the vertex $$Q$$ inside the box at rest. Neither ant can fly, but only can crawl along the face or edge of the box. Ant-1 has to reach at the vertex $$R$$ going along the edges or a face of the box from outside of the box and ant-2 has to reach vertex $$D$$ going from inside the box. Both ants start at the same time at the same speed of $$1\text{ cm/s}$$, and both follow the shortest possible route. Find the time taken by both the ants to reach their respective destinations.

In the regular pentagon $$ABCDE$$, the perpendicular at $$C$$ to $$CD$$ meets $$AB$$ at $$F$$. Prove that $$AE + AF = BE$$.

Lata wants to multiply a two-digit number with a three-digit number. When she simply placed the two-digit number to the left of the three-digit number, forming a five-digit number, it is found that this number is exactly nine times the product of the two numbers Lata took. Find the two numbers.

A grocer used a defective balance and gave $$4.8\text{ kg}$$ of almond at Rs. 4800 per kg to a customer. The customer was in doubt about the weight. So, he asked the grocer to put the almond bag in the other pan. Surprisingly it weighed only $$2.7\text{ kg}$$. The grocer apologised and charged the amount for only $$3750\text{ g}$$ (that is average of $$4.8\text{ kg}$$ and $$2.7\text{ kg}$$) of almond. In doing so, the grocer still earned the profit. Find the percentage profit of the grocer.

Let $$x$$ be a five-digit number. Let $$y$$ be the four-digit number formed from $$x$$ by deleting its middle digit. Find all such $$x$$ when $$y$$ divides $$x$$ completely.

$$A_1, A_2, A_3, \ldots, A_{20}$$ are the vertices of a regular polygon of 20 sides. $$A_1A_2P_2P_1$$ is a square on $$A_1A_2$$ interior of the polygon. $$A_3Q_3Q_4Q_5A_4$$ is the regular pentagon drawn on side $$A_3A_4$$ interior of the polygon. Prove that $$Q_3$$ lies on the line joining $$A_3P_2$$.

Suppose $$a_1, a_2, a_3, a_4, \ldots, a_n$$ is a sequence of numbers with the property that $$a_1 + a_2 + a_3 + \cdots + a_n = \frac{n(n+1)(n+2)}{6}$$, for all natural numbers $$n$$. [1] Find $$a_{100}$$, [2] Evaluate $$\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \cdots + \frac{1}{a_{89}} + \frac{1}{a_{90}}$$.

Let $$a < b < c < d < e$$ be real numbers. Among 10 sums of the pairs of these numbers, the smaller three are 32, 36, 37, while the larger two are 48 and 51. Find all possible values of $$e$$.