VTMO 2014 — Grade 7 — semifinal

$A$ is the largest four-digit number with distinct digits. Find its digit sum.

Find $t : 2014 \times 2015 = 2014 \times 2000 - 2014 + 2014 t$.

$a \equiv 1 , b \equiv 2$ modulo $8$. Find the remainder of $9 a + 3 b$.

For every integer $30$ through $50$, subtract the units digit from the tens digit. Sum all results.

Define a♣$b = \left(a + 1\right) \left(b + 1\right)$. Natural numbers $a \gt b$ satisfy a♣$b = 13$. Find a.

A sequence starts $1 , 2 , 3$; each later term is the sum of the previous three. Find term $8$.

A number increases $25 \%$. What percentage decrease returns it to the original?

A number consists of $2014$ sevens. What minimum deletions make the remaining number divisible by $63$?

Which product is a perfect square, where $n ! = 1 \times 2 \times \ldots \times n$?

Triangle angles are $\left(6 x + 5 y\right) ^{\circ} , \left(3 x + 39\right) ^{\circ} , \left(4 y + 33\right) ^{\circ}$. Find $x + y$.

In the diagram, $x = 55 ^{\circ} , y = 40 ^{\circ} , z = 35 ^{\circ}$. Find $t$.

Two angles of an isosceles triangle are $70 ^{\circ}$ and $x ^{\circ}$. Which listed value can equal $3 x$?

$AB \lt AC$, angle $B = 70 ^{\circ}$, the exterior angle at $C = 130 ^{\circ}$. $D$ lies on $AC$ with $AD = AB$. Find angle $BDC$.

Congruent equilateral triangles $ABC$ and $EBD$ are arranged as shown, with angle $ABE = 70 ^{\circ}$. Find angle $BDA$.

Rectangle $ABCD$ has perimeter $240 \,\mathrm{cm} , AB = 40 \,\mathrm{cm}$. Find the illustrated shaded area.

$A = 400 m$ track has four equally spaced points $A , B , C , D$. Start is midway along $AB$. An walks counterclockwise at $1.4 \frac{m}{s}$ for $30$ minutes. Which point is nearest?

Cutting $a = 72 \,\mathrm{cm}$ rod into $6 \,\mathrm{cm}$ pieces takes $165$ minutes. At the same cutting rate, how long to cut $a = 144 \,\mathrm{cm}$ rod into $6 \,\mathrm{cm}$ pieces?

Find the last digit of $1 ! + 2 ! + \ldots + 2014$!.

A team plays $90$ matches and wins $40$ of the first $50$. How many additional wins produce exactly $70 \%$ wins for the season?

$A , B , C , D$ lie in order on a line, with $\frac{AB}{AD} = \frac{3}{8}$ and $\frac{BC}{AD} = \frac{2}{5}$. Find $\frac{AB}{AC}$.