SEAMO 2017 — Paper C — Grade 5–6

A three-digit number equals $19$ times the sum of its digits. Find its largest possible value.

Cut a $580$ cm rope into $40$ cm and $90$ cm segments without waste. How many choices of segment counts are possible?

Define $2\oplus4=2+3+4+5=14$ and $5\oplus3=5+6+7=18$. Find $m$ if $m\oplus7=49$.

A car travels at $40$ km/h for two hours and $60$ km/h for three hours. Find its average speed.

Fill the eight circles with $1,2,\ldots,8$, each once. The corner numbers of each of the four drawn triangles sum to $12$. Find $a+b+c+d$.

Diagonals $AC,BD$ meet at $E$ and divide quadrilateral $ABCD$ into four triangles. Given $BE:DE=2:1$ and $AE:CE=1:3$, find $[ADE]:[BCE]$.

Evaluate $1-2+3-4+\cdots+2015-2016+2017$.

Melvin uses $1,2,\ldots,7$ once each to form three two-digit numbers and one one-digit number. Their sum is $100$. Find the largest two-digit number Melvin formed.

Let $\langle n\rangle$ be the smallest positive whole number not dividing $n$, e.g. $\langle7\rangle=2$, $\langle12\rangle=5$. Find $\langle\langle23\rangle\times\langle100\rangle\rangle$.

The six-digit number $\overline{abcdef}$ multiplied by $3$ gives $\overline{bcdef9}$. Find $\overline{abcdef}$.

Find the remainder on dividing the $2017$th term of $1,2,4,7,11,16,22,29,\ldots$ by $5$.

The digital clock displays one hour digit, two minute digits and two second digits, as in $7{:}20{:}15$. How many times from $7{:}00{:}00$ to $8{:}00{:}00$ have all five digits different?

How many numbers occur in both $1,3,5,7,\ldots,2015,2017$ and $1,4,7,10,\ldots,2014,2017$?

A semicircle of diameter $14$ cm and a sector of radius $14$ cm and angle $30^\circ$ overlap as shown. Find the perimeter of the shaded regions, taking $\pi=22/7$.

A camera is marked $35\%$ above its cost, then sold at a $10\%$ discount. The buyer also uses a $50$ dollar voucher. Net profit is $380$ dollars. Find the cost price.

Find the largest and smallest fractions among $9/5,12/7,27/17,36/19,54/29$, in that order.

Alan and Ben together finish a project in $35$ days. If Alan works alone for $40$ days, Ben needs a further $28$ days. If Alan works alone for $30$ days, how many further days does Ben need?

Evaluate $(\frac12+\frac13+\cdots+\frac1{2017})(1+\frac12+\cdots+\frac1{2016})-(1+\frac12+\cdots+\frac1{2017})(\frac12+\frac13+\cdots+\frac1{2016})$.

An escalator moves upwards at constant speed from floor one to floor three. Chiranjit walks at $2$ steps/s and arrives in $40$ s. Debdeep walks at $3$ steps per $2$ s and arrives in $50$ s. How many stationary steps span the escalator?

Three circles of radius $7$ cm pass through one another’s centres as shown. Find the total shaded area, taking $\pi=22/7$.