NMTC Stage 2 GAUSS Primary Level 2025

ABC is a triangle in which $$\angle CAB \colon \angle ABC \colon \angle BCA = 11 \colon 4 \colon 3$$. The line through point C making an angle $$\frac{\angle BCA}{3}$$ with BC meets the line through point B which bisects $$\angle ABC$$, at D. The bisector of $$\angle CAB$$ and the line through D making an angle $$\frac{\angle BDC}{3}$$ with CD, meet at E. Find and write the measure of $$\angle AED$$.

Consider the seven-digit number $$5a793a4$$, where a is a digit. Find all such seven-digit numbers which are divisible by 3. Write down the smallest and the largest of these numbers and write the difference between these two numbers.

N is a two-digit number. The digit 3 is affixed to the right of N to make it a three-digit number. The new number is 777 more than the original number. Find N. M is a two-digit number. M is equal to 4 times the sum of its digits. Find all such two-digit numbers. Let P be the average of such all two-digit numbers. Calculate $$N + P$$.

Look at the sequence 1, 1, 2, 3, 5, 8, 13, ... . This is called the Fibonacci sequence. Starting from the third term, each term is the sum of its immediate previous two terms. 

For Ex: 5 = 2 + 3, 8 = 3 + 5, 13 = 5 + 8 etc.

Find the remainder when the number in the 2025th term is divided by 3.

After the complete simplification of the fraction $$\frac{\left(\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\right)}{\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}\right)} + \frac{\left(\frac{1}{3}+\frac{1}{4}-\frac{1}{5}-\frac{1}{6}\right)}{\left(\frac{1}{3}-\frac{1}{4}-\frac{1}{5}+\frac{1}{6}\right)}$$, the result is of the form $$\frac{a}{b}$$, where a and b are natural numbers with no common factor other than 1. What fraction is to be subtracted from $$\frac{a}{b}$$ to get 1, and what is the integer part of this fraction?

In triangle ABC, point D is taken on the extended side AB and point E is taken on the extended side AC such that $$BC = BD = CE$$. Line-segments BE and CD are drawn, which intersect each other in F. The measure of angle BAC is $$70^\circ$$, then find the measure of angle BFD. Justify your answer.

Using the digits 0, 2, 4, 6, 8 each at least once, form and write the greatest and the smallest seven-digit number divisible by 99. Justify your answer. Also, find and write the difference between these numbers.