VTMO 2015 — Grade 8 — semifinal

Which segment is the image of $PQ$ under point reflection about $O$ in the figure?

Solve $\frac{x - 1}{x} - \frac{x}{x - 1} = 0$.

Evaluate $2015^{2} - 2011 \times 2019$.

Expand $\left(\frac{x}{3} - 2\right)^{3}$.

$ABCD$ is a parallelogram, $O$ the intersection of its diagonals, $E$ on $OD$, and $F$ the reflection of $C$ over $E$. Where must $E$ lie for $AFDE$ to be a parallelogram?

$ABC$ is right at $A , D$ lies on $BC$, and $E , F$ are its perpendicular projections on $AB , AC$. Where is $EF$ minimized?

Define $a \triangle b = 5 \left(a + b\right) - ab$; for example $8 \triangle 5 = 25$. Given $3 \triangle p = 31$, find $p$.

There are three times as many boys as girls. After four of each leave, the ratio is five boys per girl. Find the initial total.

Three unit squares stand side by side on line $d$. The middle square is rotated $45 ^{\circ}$ and placed as shown. Find the distance from its vertex $B$ to $d$.

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

$ABCD$ is a square of side $12 \,\mathrm{cm} , E$ midpoint $CD , F$ midpoint $DA$, and $G = BD \cap EF$. Find area $BCEG$.

$A = 36 \,\mathrm{cm}^{2}$ 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.

How many three-digit numbers have the tens digit equal to the mean of the hundreds and units digits?

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”?

Hai earns $\frac{5}{8}$ of Tuan salary and saves $40 \%$. Her spending is half of Tuan spending. What percentage does Tuan save?

For integers $4 \le x \le y \le 13$, how many pairs make the mean of $4 , 4 , x , y , 13$ an integer?

Choose three distinct lengths from $1 , 2 , \ldots , 10$ to form a triangle. How many choices?

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

An drives at $55 \frac{\,\mathrm{km}}{h}$ for an integer number of hours. The odometer changes from $abc$ to cba, with $a \ge 1$ and $a + b + c \le 7$. Find a $^{2} + b^{2} + c^{2}$.

There are $20$ apples, $30$ pears, $40$ bananas, and $50$ peaches. Each monkey needs whole fruits of three distinct types. What is the largest number satisfied?