SEAMO 2019 — Paper D — Grade 7–8

Given $(a+b)^3=a^3+3a^2b+3ab^2+b^3$, find the second and third terms in $(2x+3y)^3$.

Evaluate $\frac{20192018^2+1}{20192017^2+20192019^2}$.

ABCDE is a regular pentagon. O is inside it, and DOE is equilateral. Find $\angle AOC$.

Given that 2016 is a leap year and Sonia’s birthday was Sunday, June twenty-fourth, 2012, in which year will her birthday next be Sunday?

Which statement about $\sqrt5+\sqrt8$ and $\sqrt4+\sqrt{10}$ is true?

A batch of berries weighs one hundred kg, with one percent flesh and ninety-nine percent water. After drying, water is ninety-eight percent of the mass. What is the new mass?

AB and BC are diameters of semicircles ADB and CPB. AD is tangent to CPB at P, and $\angle PBC=28^\circ$. Find $\angle PAC$.

Line AB passes through ten and four on a clockface. How many minutes after 4:00 PM do the hour and minute hands make equal angles with this line?

The sum $\sum_{k=1}^{18}\frac1{k(k+1)(k+2)}$ is m/n in lowest terms. Find m+n.

Evaluate $\log\frac{25}{16}-2\log\frac57+\log\frac{32}{49}$.

Simplify $\frac{2^{n+4}-2(2^n)}{2(2^{n+2})}$.

Given $\frac c{a+b}\lt \frac a{b+c}\lt \frac b{c+a}$, which ordering is true?

Evaluate $\sqrt{14+6\sqrt5}-\sqrt{14-6\sqrt5}$.

Evaluate $\frac12+\frac24+\frac38+\frac4{16}+\cdots+\frac8{2^8}$.

Find a positive integer x such that $2^6+2^{10}+2^x$ is a perfect square.

Positive reals a,b,c satisfy $a+b+c=1$. Evaluate $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}$.

Twenty balls are numbered one through twenty. Two are chosen randomly without replacement. Find the probability that their sum is prime.

Two commercial properties cost $650000$ dollars in total. A year later the first rose twenty percent, the second fell twenty-five percent, and their values became equal. Find the first property’s original value.

In sector OAB, the source writes $\widehat{AB}=18$ cm. OA and OB are tangent to two mutually tangent circles. The circle centered at C has radius six cm. Find the radius of the circle centered at D.

Square ABCD has side $2\sqrt{15}$. E,F are midpoints of AB,BC. AF meets DE and DB at M,N. Find $[DMN]$.