VTMO 2026 - Grade 8 - Final - SET 5 - Self-study

Let $A=(a+b)(a+2b)(a+2015b)(a+2016b)+1007^2b^4$. Prove that $A$ is a perfect square for every $a,b\in\mathbb Z$.

To decide which flowers to plant along the city's central road, residents of Ho Chi Minh City were surveyed. There were $N$ respondents. Exactly $9/14$ of them considered flower colour important, and exactly $7/12$ considered fragrance important. In total, $753$ considered both factors important. Find the smallest possible value of $N$.

Let $a,b,c$ be pairwise distinct. Simplify $$A=\left(\frac1{(a-b)^2}+\frac1{(b-c)^2}+\frac1{(c-a)^2}\right):\left(\frac1{a-b}+\frac1{b-c}+\frac1{c-a}\right)^2.$$

Find all integer pairs $(x,y)$ satisfying $2x^2-xy+3x-2y=5$.

Let $a,b,c,d$ be positive integers with $a>b$ and $c>d$. Prove that if $a+b+c+d=ab-cd$, then $a+c$ is composite.