SEAMO 2023 — Paper D — Grade 7–8

Evaluate $\frac{2023^2+2123^2-2(2023)(2123)}{25}$.

The number minus one is a root of $x^2+ax+b=0$, and $(a-b)/(a+b)=1/7$. Find ab.

How many triangles are in the figure?

Square ABCD has side ten cm. E,F are midpoints of BC,AB, and G is the intersection of DE and FC. Find triangle GDC’s area.

Evaluate $\sum_{k=2}^{18}1/[k(k+1)(k+2)]$.

Different letters represent different digits in $GNAT+GNAT=TAKEN$. Find A.

Find the least real k for which $\sqrt{x-2}+\sqrt{7-x}\le k$ for every real x where the radicals are defined.

Two concentric circles bound a shaded annulus. A chord of the larger circle has length $\sqrt{20}$ cm and is tangent to the smaller circle. Find the annulus area.

Evaluate $1+1/3+1/3^2+\cdots+1/3^{10}$.

For rational m, find the least value of $|m-1|+|m-3|+|m+5|+|m+6|$.

Simplify $\tan1^\circ\tan2^\circ\cdots\tan89^\circ$.

Find the remainder when $1021^{1022}$ is divided by $1023$.

For an equilateral triangle, its circumcircle-to-incircle area ratio is m:n in lowest terms. Find m+n.

Evaluate $\sum_{k=1}^{10}k/2^k$.

Towns A,B are thirty-six km apart. Alan and Betty start simultaneously from opposite towns. After meeting Alan needs two hours thirty minutes to finish, Betty one hour thirty-six minutes. Find Betty’s speed in km/h.

Primes p,q,r satisfy $pqr=11(p+q+r)$. Find the least $p^2+q^2+r^2$.

For $x\gt 0$, find the minimum of $x+4/x$.

Evaluate $\sqrt{2023+2022\sqrt{2023+2022\sqrt{2023+2022\sqrt{\cdots}}}}$.

Integers x,y satisfy $323x+457y=1103$, $177x+543y=897$. Find $x^4+4x^2y^2+5y^4$.

ABC is right-angled with hypotenuse BC and $AC=5$. D is midpoint of BC, E lies on AC, and $DE=AE=2$. Find AB.