VTMO 2026 - Grade 7 - Final - SET 1 - Self-study

Two motorcyclists start simultaneously from A and B towards each other. The first reaches B and turns back; the second reaches A and turns back immediately. Their first meeting is 15 km from A and their second is 9 km from B, with opposite directions at the second meeting. Find AB.

a) Suppose $\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}$ with $abc\ne 0$. Prove $x: y: z=a: b: c$.

b) A six-digit number $\overline{abcdef}$ is divisible by 7. Prove that the six-digit number $\overline{efabcd}$ is also divisible by 7.

Oil tanker B is 100 km due north of cargo ship A. Cargo ship A travels northwest at 20 km/h, while B travels southeast at 30 km/h. How many kilometres has the cargo ship travelled when the two ships are closest?

A number is self-descriptive if its nth digit counts the occurrences of digit n−1 in that number. For example 21200 is self-descriptive: its initial 2 counts two zeros, the next 1 counts one 1, the next 2 counts two 2 s, and the final two zeros indicate no 3 s or 4 s. Find all four-digit self-descriptive numbers.

A teacher writes a two-digit prime and a one-digit prime. An and Binh calculate their sum. An’s calculator works normally; Binh’s faulty digit keys replace each entered digit d by 9−d:0→9,1→8,2→7,3→6,4→5,5→4,6→3,7→2,8→1,9→0. Both calculators display the same sum. Find the two primes.