VTMO 2020 — Grade 6 — vtmo

Fahrenheit and Celsius satisfy $F = 9 \frac{C}{5} + 32$. Convert $77 ^{\circ} F$ to $^{\circ} C$.

A digital clock displays $HH : MM$ from $00 : 00$ to $23 : 59$. How many daily displays contain exactly two $2 s$ and two $0 s$?

Use $2 , 3 , 4 , 5 , 6 , 7$ once each to form two numbers. Find their least positive difference.

De Morgan $\left(1806 \text{--} 1871\right)$ says“I $am x$ years old in year $x^{2}$”. Find $x$.

Write eight consecutive natural numbers on a cube’s eight vertices. What maximum number of edges have an odd sum of endpoints?

Divide a two-digit number by its digit sum. Find the largest possible remainder.

An empty $4000$-liter tank fills at $20$ liters/minute. After $20$ minutes it leaks $2$ liters/minute while pumping continues. Find total filling time.

Split $70$ candies among six students with distinct positive counts. What is the minimum possible largest count?

An completes $n$ problems with some wrong. Correcting two wrong answers makes $76 \%$ correct; discarding those two makes $75 \%$ correct. Find $n$.

Use integers $1 \text{--} 32$ once each to form $16$ fractions. What maximum number can be integers?

Buying a watch at $40 \%$ off leaves mother $16$; at full price she would be short $4$. Find its full price. The source English paragraph instead prints $30 \%$, conflicting with VI $40 \%$.

The pictured $T$-shaped structure comprises equal small cubes. How many single-cube removals preserve its surface area?

Find the digit sum of $999999999^{2}$, where the base has nine $9 s$.

A runner and cyclist start together around a lake in the same direction. The cyclist is seven times faster. Mark every meeting position. How many distinct positions occur?

Find the smallest prime that can be expressed as a sum of two primes, and also as sums of three, four, and five distinct primes.

$A = 14 \,\mathrm{cm}$ square contains an inscribed circle, which contains an inscribed smaller square as pictured. Find the shaded area outside the smaller square.

Fill $a = 4 \times 5$ grid with $0 , 1 , 2$ so each row and column sum is divisible by $3$. What maximum number of $1 s$ is possible?

Five teams play a single round robin: win $3$, draw $1$, loss $0$. At some point all five have distinct positive scores. What minimum number of matches has been played?

Vinh and Khang know only their own scores, natural numbers at most $20$. Vinh:“Our product is surely even.” Khang:“It is surely divisible by $20$.” Vinh:“Our scores differ and sum above $20$.” Khang:“I cannot determine yours, but our average exceeds mine.” Vinh:“I know yours, and our difference is divisible by $3$.” Khang:“Now I know yours.” Find Vinh’s score.

Nine congruent rectangles are arranged as illustrated within a square of side $23$. Find one rectangle’s perimeter.