VTMO 2022 — Grade 8 — vtmo

Question 1

Evaluate the product of $\frac{k^{3} + 1}{\left(k - 1\right)^{3} - 1}$ for $k = 3 , \ldots , 10$.

$150$
$155$
$160$
$165$
$170$

Question 2

An, Binh, Chau leave A toward $B$ at $6 , 7 , 8 AM$ with speeds $20 , 30 , 40 \frac{\,\mathrm{km}}{h}$. When is Chau midway between the others?

$9 : 00$
$9 : 20$
$10 : 00$
$10 : 20$
$11 : 00$

Question 3

$P$ has remainders $- 4$ on division by $2 x + 1$ and $5$ by $5 x - 2$; division by $10 x^{2} + x - 2$ has quotient $x$ and linear remainder. Find $P \left(2\right)$.

$101$
$92$
$61$
$181$
$52$

Question 4

A trapezoid midline divides its area in ratio $2 : 1$. Find the ratio of the larger to smaller base.

$5 : 1$
$4 : 1$
$3 : 1$
$5 : 2$
$3 : 2$

Question 5

How many right triangles can be formed from the vertices of a regular $22$-gon?

$242$
$200$
$132$
$110$
$220$

Question 6

Seven candy bags contain $10 , 15 , 20 , 25 , 30 , 35 , 40$. Choose six and give equal totals to two distinct children without opening bags. Count distributions.

$4$
$5$
$6$
$8$
$10$

Question 7

$\left[a\right] = \operatorname{floor} \left(a\right)$. How many integers $n$ make $| \operatorname{floor} \left(\frac{5 n^{2} + 9 n - 1}{2}\right) |$ prime?

Question 8

Cubes of sides $x \gt y$ total $13 m^{3}$. Square-base cuboids with bases $x , y$ and respective heights $y , x$ total $4 m^{3}$, as pictured. Find $\frac{x}{y}$.

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Question 9

$B$ lies on $AC$. Equilateral $ABD , BCE$ lie on the same side. $N$ on $BE$ has $NE : NB = 4 : 5$; $AN$ meets $BD , CE$ at $M , P$. $AB = 16 \,\mathrm{cm}$ and areas $AMB , ENP$ are equal. Find $BC$.

$40$
$45$
$48$
$50$
$56$

Question 10

Square $ABCD$ has side $3 \,\mathrm{cm} , M , N , P , Q$ on $AB , BC , CD , DA , AM = BN = CP = DQ = 1$. $E , F$ are $BD$’s intersections with $MN , PQ$. Find $EF$.

Question illustration

$3 \frac{\sqrt{3}}{2}$
$\sqrt{3}$
$\sqrt{2}$
$4 \frac{\sqrt{2}}{3}$
$5 \frac{\sqrt{2}}{3}$

Question 11

Repeatedly cut one piece of a hexagon by a straight line avoiding vertices. The final pieces total $90$ sides. How many cuts?

Question 12

For integer $x$, minimize $P = \frac{4 x^{4} + 81}{2 x^{2} + 6 x + 9}$.

Three equal candy boxes are divided into three type $- I$ bags, type $- II$ bags, and type $- III$ bags. One type $- I$ bag contains the combined contents of one type $- II$ and one type $- III$ bag, and all three bag sizes differ. How many bags total?

Question 14

Exactly three of four claims hold: $A - 4$ prime; $A - 5$ square; $4 A - 3$ square; $4 A + 12$ has exactly six positive divisors. Find $A$.

Question 15

$ABC$ is right isosceles at $A , M$ inside, $MAC : MBA = 1 : 2 , AB = 41 \,\mathrm{cm} , MA = 18 \,\mathrm{cm}$. Find area $MAB$.

Question 16

$ABC$ fixed, $D$ arbitrary. On rays $DB , DC$ choose $E , F$ with $DE = 2 DB , DF = 3 DC$. Construct parallelogram $ADEM , N$ midpoint $MF$. Prove $DN$ passes through a fixed point.

Question 17

In $a = 5 \times 5$ grid, touching a cell toggles it and its edge-neighbors. Black means off, white on. Initially black cells are $\left(1 , 1\right) , \left(1 , 3\right) , \left(1 , 5\right) , \left(5 , 1\right) , \left(5 , 5\right)$, as shown. Can all lights be switched off?

Question illustration