SEAMO 2017 — Paper D — Grade 7–8

Solve $\frac x{3+x}-\frac x{4-x}=2$.

Find all positive values of $n$ for which $2^n-1$ is divisible by seven.

Coffee types A and B cost forty and sixty dollars per kg and are mixed in mass ratio $m:n$. A’s price decreases $15\%$, B’s increases $15\%$, while mixture cost remains unchanged. Find $m:n$.

Evaluate $\frac{400^2(254^2+246^2)(254^4+246^4)}{254^8-246^8}$.

Evaluate $\frac{20172016^2}{20172015^2+20172017^2-2}$.

Given $m=\frac a{b+c}=\frac b{a+c}=\frac c{a+b}$ and $a+b+c\ne0$, find $m$.

A rectangle is inscribed in a square as shown. The four isosceles right triangles outside it have total area $98\text{ cm}^2$. Find its diagonal $XY$.

Evaluate $\left(\frac23+\frac34+\cdots+\frac{49}{50}\right)\left(\frac12+\frac23+\cdots+\frac{48}{49}\right)-\left(\frac12+\frac23+\cdots+\frac{49}{50}\right)\left(\frac23+\frac34+\cdots+\frac{48}{49}\right)$.

Evaluate $\sqrt{2017+2016\sqrt{2017+2016\sqrt{2017+2016\sqrt{\cdots}}}}$.

Given $a+1/a=5$, find $a^4+1/a^4$.

Evaluate $\log_{1/2}8+\log_2 64-\log_5(1/125)$.

Rectangle $ABCD$ has $AB=6,BC=4$. E is the midpoint of AB, and $DF\perp CE$. Find $DF$.

Find the smallest number leaving remainders $33,59,98$ on division by $52,78,117$, respectively.

Triangle ABC has area $8\text{ cm}^2$, $AE=DE$, $BD=2CD$. In the shown configuration, find the total shaded area BDE and AEF.

In triangle ABC, $AN=BM=AB$, with N on BC and M on AC. $AN,BM$ meet at P, and $\angle C=38^\circ$. Find $\angle APB$.

For a nonzero real number x, let $p=x-1/x$, $q=x^2-1/x^2$. Express $p^4+q^2+4p^2$ in terms of q.

A motorboat takes six hours from port A to B on the same river bank, and eight hours returning. Current speed is $2.5$ km/h. Find its speed in still water.

Find the ratio of the incircle area to the circumcircle area of an equilateral triangle.

Arrange $3^{50},4^{40},5^{30}$ in ascending order.

Sara selects two oranges from fifteen: ten good and five bad. The probability of at least one good is $m/n$. Find $m+n$.