VTMO 2021 — Grade 6 — cup

Place $1 \text{--} 20$ once around twenty pictured circles. $A$ and $B$ are the largest and smallest adjacent sums. Minimize $C = A - B$ and exhibit a placement.

$a$)Positive integers $a , b$ satisfy $56 a = 65 b$. Prove $a + b$ composite. $b ) A = 14 \,\mathrm{cm}$ square contains the pictured two quarter-circle arcs. Find the positive difference of areas $\left(I\right) , \left(II\right)$, using $\pi = \frac{22}{7}$.

What maximum number can be selected from $1 \text{--} 162$ without one being twice another?

Duc and Thao have candies in ratio $4 : 5$. Duc receives one third extra of his amount, Thao eats one third of hers, and their total drops by two. Find the original total.

Exactly one of $A , B , C , D$ stole gold. A says“$B$ is a liar”and“$C$ or $D$ stole it”; $B$“$C$ is a liar”and“$D$ or A stole it”; $C$“$D$ is a liar”and“$A$ or $B$ stole it”; $D$“$A$ is a liar”and“$B$ or $C$ stole it”. A truthful person has two true statements; a liar at least one false statement. How many of the eight statements are true?

Eighteen boys and $23$ girls plant trees. Each individual plants one. An all-boy group $n$ plants $n + \operatorname{floor} \left(\frac{n + 1}{4}\right)$; an all-girl group $k$ plants $k + \operatorname{floor} \left(\frac{k + 1}{6}\right)$; a mixed group $p , q \gt 0$ plants $p + q + \operatorname{floor} \left(\frac{p + q + 2}{5}\right)$. Find the maximum by combining group plans. For example $\operatorname{floor} \left(\frac{5}{3}\right) = 1 , \operatorname{floor} \left(\frac{10}{5}\right) = 2$.