VTMO 2015 — Grade 7 — semifinal

Find the last digit of $2015^{0} + 2015^{1} + \ldots + 2015^{15}$.

A handspan is $2 dm$. A rectangular room measures $22$ handspans by $18$. Find its area in $dm^{2}$.

The diagram has exterior apex angle $140 ^{\circ}$ and left base angle $80 ^{\circ}$. Find $x$, vertically opposite the right base angle.

Distinct letters represent distinct digits: $77 P + 6 QP + QQP = 1 PP = 7$. Find $P + Q$.

A square has the same perimeter as the right triangle with legs $8 , 6 \,\mathrm{cm}$ shown. Find the square area.

a cows yield $b$ liters in $c$ days at a constant rate. How many liters do $d$ cows yield in $e$ days?

Define $a \triangle b = 5 \left(a + b\right) - ab$. If $3 \triangle p = 31$, find $p$.

Three consecutive odd numbers have product $9177$. Find their sum.

Positive integers are arranged in triangular rows: row $1$ has one entry, row $2$ two, and so on as shown. Which row contains $2015$?

A box has $400$ green, white, red, yellow, orange balls. Green: white: red $= 1 : 2 : 4$ and white: yellow: orange $= 1 : 3 : 6$. What minimum blind draws guarantee $50$ of one color?

$A = 36 \,\mathrm{cm}^{2}$ square is partitioned into four congruent rectangles and an inner square as shown. The inner square perimeter equals one rectangle perimeter. Find each rectangle area; diagram not to scale.

Shop A sells a bicycle for $200$ dollars. $B$ charges $15 \%$ more, but gives the customer $10 \%$ off. What is paid?

Binh answers $80 \%$ of ten multiple-choice questions and $70 \%$ of thirty fill-in questions correctly. Find the overall correct percentage.

Square $PQRS$ has $M , N$ the midpoints of $PS , RS$. Area $\left(SMN\right) = 18 \,\mathrm{cm}^{2}$. Find area $\left(QMN\right)$.

How many three-digit numbers have adjacent digits differing by $3$, such as $141$ and $258$?

$PQR$ is right isosceles at $Q$. $QS , TU , VW$ are perpendicular to $PR$, and $ST , UV$ perpendicular to $QR$ as shown. Find shaded area divided by area $PQR$.

Integers $4 , 4 , x , y , 13$ satisfy $4 \le x \le y \le 13$. How many pairs give an integer average?

Find the greatest natural $k$ such that $k^{2}$ divides $10$!.

Hot water fills a tank in $23$ minutes, cold in $17$. Start hot first. When should cold be opened so the final hot volume is $1.5$ times the cold volume?

Chau lies Friday, Saturday, Sunday and is truthful otherwise. Minh is truthful Tuesday, Wednesday, Thursday and lies otherwise. When can both say“I will lie tomorrow”?