SEAMO X 2023 — Paper D — Grade 7–8

Define $a*b=ab-a-b$. If $2*x=1$, find x.

Evaluate $44+45+46+\cdots+2022+2023$.

Which is largest: $4^{75},3^{120},2^{125}$?

Given $2x_1+x_2+x_3+x_4=5$, $x_1+2x_2+x_3+x_4=10$, $x_1+x_2+2x_3+x_4=20$, $x_1+x_2+x_3+2x_4=40$, find $155x_3+49x_4$.

Write $1/2+3/4+7/8+\cdots+1023/1024$ as the simplest mixed number $w\frac mn$. Find w+m+n.

Given $\log(11/15)+\log(640/297)-2\log(8/9)=\log x$, find x.

Square ABCD has area forty-five square cm. M is midpoint of AD and G is the intersection of BM and AC. Find the total shaded area.

How many numbers must be selected from one through twelve to guarantee a pair whose difference is six?

The sum of the first n terms of a sequence is $2n^2+3n$. Find its fourth term.

Find the least integer n such that $(n-10)/(9n+13)$ is a nonzero reducible fraction.

If $a^3+b^3=1343$ and $a+b=17$, find $(a-b)^2$.

Adding one hundred seventeen grams of water to a saline solution makes its salt concentration twenty percent. Adding a further one hundred seventeen grams of salt makes the concentration $33\frac13$ percent. Find the original solution mass.

Find the last four digits of $7^{144}$.

In the right triangle ABC shown, E,D are midpoints of AC,BC. Given $EB=41$ cm and $AD=38$ cm, find AB.

Distinct digits a,b,c satisfy $\overline{acb}+\overline{bac}+\overline{bca}+\overline{cab}+\overline{cba}=3194$. Find $\overline{abc}$.