SEAMO 2026 Paper E SET 1 – Self-practice

Let a, b, c and d be integers such that $(a^2+b^2)(c^2+d^2)=29$. Find the value of $a^2+b^2+c^2+d^2$.

Find the value of $\frac{300^2}{253^2-247^2}$.

It is known that there is only one pair of positive integers a and b such that $a\le b$ and $a^2+b^2+3ab=719$. Find $(a+b)$.

In an equilateral triangle $\triangle ABC$, D and E are two points on AB and AC, respectively. BE and CD intersect at F. It is known that $\angle BFC=120^\circ$. Which of the following is TRUE about the magnitudes of AD and CE?

How many consecutive zeros are there at the end of the decimal representation of 80!?

Find the remainder when $6^{273}+8^{273}$ is divided by 49.

The infinite sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15… is obtained by writing the positive integers in order. What is the $2022^{\mathrm{nd}}$ digit in the sequence?

Which of the following numbers is the smallest?

Let n! denote the product $n\times(n-1)\times(n-2)\times\cdots\times2\times1$. If $f(n)=\frac{(100/3)^n}{n!}$, find the positive integer n for which f(n) is largest.

$\triangle ABC$ is an equilateral triangle. D is a point outside $\triangle ABC$ such that AC=AD. Given that $\angle CDB=x$, find the value of x.

If $A=40^\circ$ and $B=5^\circ$, find the value of $(1+\tan A)(1+\tan B)$.

If $a+b=2$ and $a^2+b^2=5$, find the value of $a^3+b^3$.

Simplify $\sqrt{3+2\sqrt2}+\sqrt{3-2\sqrt2}$.

If n and k are positive integers that satisfy $\frac7{13}<\frac n{n+k}<\frac6{11}$, find the smallest n that satisfies the inequality.

Write down the last four digits of $7^{128}$.

In a pack of 10 watches,3 are known to be defective. If 2 watches are selected at random from the pack, what is the probability that at least one is defective?

Evaluate $40^\circ+2\tan10^\circ$.

Evaluate $\log_5\sqrt{5\sqrt{5\sqrt{5\cdots}}}$.

In a survey concerning the brands of coffee for consumers, it was found that 50% drink Brand A,45% drink Brand B,40% drink Brand C,25% drink both brands A and B,10% drink both brands B and C,16% drink both brands A and C and 8% drink all 3 brands. What percentage of the consumers surveyed do not drink coffee?

Simplify $\sqrt{\log_2 3\times\log_2 12\times\log_2 48\times\log_2 192+16}-\log_2 12\times\log_2 48+10$.