SEAMO 2023 — Paper E — Grade 9–10

Evaluate $\dfrac{\sqrt{40}+6}{\sqrt{40}-6}+\dfrac{\sqrt{40}-6}{\sqrt{40}+6}$.

Given $\sin\alpha+\cos\alpha=\sqrt2$, find $\sin\alpha\cos\alpha$.

For a positive integer $a$, both $100a+64$ and $201a+64$ are perfect squares less than $10000$. Find $a$.

Let $A=1\cdot2\cdot3\cdots2022\cdot2023$. How many consecutive zeros occur at the end of $A$?

In square $ABCD$, the diagonals meet at $O$. The bisector $AF$ of $\angle BAC$ meets $BD$ at $G$ and $BC$ at $F$. Write $OG:CF=m:n$ in coprime positive integers and find $m+n$.

Evaluate $\displaystyle\sum_{r=1}^{89}\log_{10}\tan\frac{\pi r}{180}$.

Find the largest positive integer $n$ such that $2^n$ divides $3^{1024}-1$.

In triangle $ABC$, $D$ lies on $BC$, $AD$ bisects $\angle BAC$, and $E$ is the foot of the perpendicular from $C$ to $AB$. Given $\angle CDA=45^\circ$, find $\angle BED$.

Simplify $\bigl||x-1|-2\bigr|+|x+1|$ for $1\le x\le3$.

Given $\log_2[\log_{1/2}(\log_2 x)]=\log_3[\log_{1/3}(\log_3 y)]=\log_5[\log_{1/5}(\log_5 z)]=0$, which relation is true?

For positive real numbers $a,b,c$, find the minimum of $u=\left\lfloor\dfrac{a+b}{c}\right\rfloor+\left\lfloor\dfrac{b+c}{a}\right\rfloor+\left\lfloor\dfrac{c+a}{b}\right\rfloor$, where the brackets denote the integer part.

Triangle $ABC$ has $AB=5$, $BC=8$, $CA=11$. Point $D$ lies on $BC$ and $AD$ bisects $\angle BAC$. Find the area of triangle $ACD$.

Let $S=1+\dfrac1{2^2}+\dfrac1{3^2}+\cdots+\dfrac1{2023^2}$. Find $\lfloor S\rfloor$.

The first $n$ terms of $\{a_n\}$ sum to $S_n=2a_n-1$. A sequence $\{b_n\}$ satisfies $b_1=3$ and $b_{k+1}=b_k+a_k$ for $k\ge1$. Find the sum of the first $8$ terms of $\{b_n\}$.

Join all intersections of the graphs $x^2+(y-1)^2=1$ and $9x^2+(y+1)^2=9$. What figure is formed?

Noppon travels from $A$ to $B$ and back. He goes downhill at $12$ km/h and uphill at $4$ km/h. The outward trip takes $50$ minutes and the return takes $1\frac12$ hours. Find the distance between $A$ and $B$ in km.

A positive integer $n$ makes $n^2+5n+13$ a perfect square. Find $n$.

In triangle $ABC$, $a,b,c$ are opposite angles $A,B,C$ and $\dfrac{b+c}{10}=\dfrac{c+a}{15}=\dfrac{a+b}{11}$. Find $\sin A:\sin B:\sin C$.

In the figure, $\angle A=\angle BEF=\angle EBC=\angle ECD=90^\circ$, $\angle ABF=30^\circ$, $\angle BFE=45^\circ$, $\angle ECB=60^\circ$, and $AB=2CD$. Find $\tan\angle CDE$.

$I$ is the incentre of triangle $ABC$. The extension of $AI$ meets the circumcircle at $D$. Given $AB=3$, $AC=4$ and $[IBC]=[DBC]$, find $BC$.