VTMO 2016 — Grade 9 — semifinal

Positive integers $a , b$ satisfy $\frac{1}{a} + \frac{2}{b} = \frac{8}{2 a + b}$ and $2010 \lt 4 a + 7 b \lt 2016$. Find $a + b$. Original choices: $A = 336 , B = 333 , C = 330 , D = 339 , E = 342$.

Find the largest $n$ for which $2016^{n}$ divides $59$!.

Find the units digit of $1 \times 3 \times 5 \times 7 + 3 \times 5 \times 7 \times 9 + \ldots + 2011 \times 2013 \times 2015 \times 2017$.

A bus has one-person and two-person seats. Once it carries $13$ people with $9$ seats empty; another time $10$ people with $6$ seats empty. How many seats does it have, excluding the driver?

The four sides and one diagonal of a quadrilateral have lengths $1 , 2 , 2.8 , 5 , 7.5$. Which is the diagonal?

How many triangles have integer sides $x \lt y \lt z$ and perimeter $27 \,\mathrm{cm}$?

$A \left(0 , a\right) , D \left(0 , - 1\right) , B \left(2 , - 1\right) , C \left(3 , 2\right) , O \left(0 , 0\right)$. Triangles $ADB$ and $CBO$ have equal areas. Find a as shown.

An starts $100 m$ west of Binh, Chau $160 m$ east. An walks south $20 m /$ min, Chau north $41 m /$ min, Binh stays still. When is Chau–Binh twice An–Binh?

How many odd numbers must be selected from $\{ 123 , 124 , \ldots , 2016 \}$ to guarantee a pair summing to $2016$?

A quarter-circle of radius $R$ contains two tangent semicircles and a circle tangent to both, as pictured. The shaded area is $\pi \frac{R^{2}}{x}$. Find $x$.

A product of five distinct odd primes is the five-digit number abcab with $c = 0$. How many prime sets are possible?

Distribute $100$ apples into $20$ bags, each containing exactly $3$ or $x$ apples. How many integer $x$ values are possible?

Right triangle $P _{1} QR _{1}$ has $QP _{1} 2 \,\mathrm{cm} , QR _{1} 5 \,\mathrm{cm}$. Extend each ray with successive points $1 \,\mathrm{cm}$ apart. Find $n$ such that area $P _{n} P _{n}$₊$_{1} R _{n}$₊$_{1} R _{n}$ is $2016 \,\mathrm{cm}^{2}$.

Notebooks cost $10000 VND$. A offers one free per five purchased (nine purchased earns one). $B$ discounts $15 \%$ on purchases of at least four. What minimum cost buys $14$ notebooks, allowing both stores?

Luhn algorithm: reverse the digits; sum odd-position digits as $A$; double even-position digits and sum digits of their products as $B$. Validity requires $A + B$ divisible by $10$. Example $4634 8932 1298 2767$ reverses to $7672 8921 2398 4364$ with $A = 45 , B = 45$. The valid string $2853 00 QP P = 003 6002$ has digits $P \le Q$. How many pairs $\left(P , Q\right)$?