SEAMO X 2024 — Paper D — Grade 7–8

Find the remainder when $2024^{2024}$ is divided by seven.

Triangle ABC has $AB=3,BC=4,AC=5$. If $\sin2A=x/y$ in lowest positive terms, find x+y.

Find the remainder when $x^3+3x^2+2x+1$ is divided by $x-2$.

Evaluate $\sqrt[256]{(2+1)(2^2+1)(2^4+1)(2^8+1)\cdots(2^{256}+1)+1}$.

For $a\gt 1$, find the least value of $2\log_{100}a-\log_a(0.0001)$.

The four-digit number $\overline{abcd}$ is divisible by eleven, $b+c=a$, and $\overline{bc}$ is a perfect square. No digit is zero. Find the least number.

Given $y=\sqrt{\frac{x^2-2}{5x-4}}-\sqrt{\frac{x^2-2}{4-5x}}+2$, find $x^2+y^2$ for real x,y.

For positive integers n, let $A_n=(20^n+24^n)/n!$. Find n maximizing A_n.

Distribute seven differently colored beads into four identical baskets, none empty. How many ways are there?

Evaluate $480\sum_{k=2}^{15}1/(k^2-1)$.

The quadratic $x^2+2(k+3)x+k^2+3=0$ has distinct real roots $\alpha,\beta$, and k is integral. Find the minimum of $(\alpha-1)^2+(\beta-1)^2$.

Real a,b,c satisfy $a+2b+c=4$. Find the maximum of ab+bc+ca.

ABCD is a square of side two cm. BPC is equilateral. If triangle BPD has area k square cm, find $\sqrt3-k$. Use the source diagram.

A cube has volume one cubic cm. M is midpoint of edge GC. Its shortest surface path from A to M has length $\sqrt a/b$ cm, with integral a,b and square-free a. Find a+b.

I is the incenter of triangle ABC. AI extended meets the circumcircle at D. Given $AB=3,AC=4$ and equal areas of IBC and DBC, find $4BC$.