VTMO 2026 - Grade 7 - Final - SET 10 - Self-study

Prove that $A=\dfrac12+\dfrac2{2^2}+\dfrac3{2^3}+\cdots+\dfrac{2014}{2^{2014}}\lt 2$.

Partition the integers from 1 to 23 into two groups, each containing at least two numbers. How many different partitions make the products of the two groups relatively prime?

Do positive integers $a,b,k$ exist such that $\gcd(a,b)=k(a-b)$ and $\operatorname{lcm}(a,b)=k(a+b)$ simultaneously?

Working alone, team A needs 20 days to finish a project and team B needs 30 days. They undertake it together, but A rests for 3 days and B for 2 days, with no day on which both rest. How many days does it take to finish?

In triangle $ABC$, $AB\lt AC$. Choose $E$ on the ray opposite to $BA$ with $BE=AC$, and $F$ on the ray opposite to $CA$ with $CF=AB$. Let $M$ be the midpoint of $BC$, $N=AM\cap EF$, and $D=NC\cap EM$.

a) Prove $MA=MN$.

b) Prove $AN=DF$.