SEAMO 2022 — Paper E — Grade 9–10

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

Evaluate $\dfrac{300^2}{253^2-247^2}$.

There is only one pair of positive integers $a,b$ with $a\le b$ and $a^2+b^2+3ab=719$. Find $a+b$.

In equilateral triangle $ABC$, $D$ lies on $AB$ and $E$ on $AC$. Lines $BE$ and $CD$ meet at $F$, with $\angle BFC=120^\circ$. Which relation between $AD$ and $CE$ is true?

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

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

Writing the positive integers consecutively gives $123456789101112131415\ldots$. What is the $2022$nd digit?

Which of the following numbers is the smallest?

Let $n!=n(n-1)\cdots2\cdot1$ and $f(n)=\dfrac{(100/3)^n}{n!}$. For which positive integer $n$ is $f(n)$ largest?

$ABC$ is an equilateral triangle. Point $D$ lies outside the triangle and $AC=AD$. Given $\angle CDB=x$, find $x$.

If $A=40^\circ$ and $B=5^\circ$, evaluate $(1+\tan A)(1+\tan B)$.

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

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

Positive integers $n,k$ satisfy $\dfrac7{13}\lt \dfrac n{n+k}\lt \dfrac6{11}$. Find the smallest $n$.

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

Of $10$ watches, $3$ are defective. Two watches are selected at random without replacement. What is the probability that at least one is defective?

Evaluate $\tan40^\circ+2\tan10^\circ$.

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

In a survey, $50\%$ drink coffee brand $A$, $45\%$ brand $B$, and $40\%$ brand $C$. Also $25\%$ drink both $A,B$, $10\%$ both $B,C$, $16\%$ both $A,C$, and $8\%$ all three. What percentage do not drink coffee?

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