VTMO 2015 — Grade 9 — semifinal

Which pictured segment is the reflection of $PQ$ in the $x$-axis?

Evaluate $\sqrt{4 - 2 \sqrt{3}} + \sqrt{7 - 4 \sqrt{3}}$.

$f \left(x\right) = x + \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}$. Find $f \left(2015\right)$.

Worker $1$ has made $12$ items and makes $2$ per hour. Worker $2$ has made $6$ and makes $5$ per $2$ hours. When their totals coincide, how many has each made?

A circle is divided into $12$ equal arcs labeled A through $L$. In the picture $x = \angle EAG$ and $y = \angle AGI$. Find $x + y$.

$A = 3 \times 4$ rectangle has a vertex $\left(1 , 2\right)$. Which point cannot be another vertex?

The table lists $x = - 2 , - 1 , 0 , 1 , 2$; $f \left(x\right) = - 5 , - 2 , - 3 , 7 , 4$; $g \left(x\right) = 7 , 6 , 8 , 3 , 4$. Define $h \left(x\right) = 2 f \left(x\right) + g \left(2 x\right)$. Find $h \left(1\right)$.

There are two a values for which $4 x^{2} + ax + 8 x + 9 = 0$ has a double root. Find their sum.

In triangle $ABC , \angle B = 60 ^{\circ} , BC = 8 \,\mathrm{cm}$ and $AB + AC = 12 \,\mathrm{cm}$. Find $AB$.

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

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

For distinct positive $a , b$, which expression is largest?

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

Four circles have centers on $AE$ and share tangency at $E$. $D$ is the largest center, and $AB = BC = CD = 1$. Find the shaded area between the circles with diameters $BE$ and $CE$.

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

A circle has diameter $AB$, with $C$ on $AB$ and $BC = 2 AC$. Let $D$ lie on the circle with $DC \perp AB$, and $DE$ be another diameter. Find area $\left(CDE\right) /$ area $\left(ADB\right)$.

Arrange $1 , 2 , 3 , \ldots$ in left-aligned rows containing $1 , 2 , 3 , \ldots$ numbers, as shown. How many smaller numbers lie in the column containing $2000$?

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 $^{2} + b^{2} + c^{2}$.

Ashley, Bethany, Caitlin have two-digit prime jersey numbers. Ashley+Caitlin is Bethany’s birthday, already past; Bethany+Caitlin is today; Ashley+Bethany is Caitlin’s upcoming birthday in the same month. What is Caitlin’s number?

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?