SEAMO 2021 — Paper E — Grade 9–10

Evaluate $\left(1+\frac12\right)\left(1+\frac1{2^2}\right)\left(1+\frac1{2^4}\right)\cdots\left(1+\frac1{2^{1024}}\right)$.

How many positive integers $n$ satisfy $3^{200}\lt n^{100}\lt (123n)^{50}$?

Let $a_n$ be the last two digits of $6^n$ for positive integer $n$; e.g. $a_1=06,a_2=36,a_3=16$. Find the last two digits of $a_1+\cdots+a_{2021}$.

$ABCD$ is a rectangle. Points $X,Y$ lie on $BC,CD$, and the areas of $\triangle ABX,\triangle CXY,\triangle AYD$ are $3,4,5$. Find $[\triangle AXY]$.

Find $a$ such that $\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}=7$.

Find the coefficient of $x^{49}$ in $(x+1)(x+2)\cdots(x+50)$.

In $\triangle ABC$, $\angle A=30^\circ$, $\angle C=90^\circ$, $AC=1$. Construct equilateral triangles $BCP,CAQ,ABR$ externally. Let $T=QR\cap AB$. Find $[PQT]$.

Find the least positive integer $n$ for which $n/5$ is a perfect square, $n/2$ is a perfect cube, and $27\mid n$.

Evaluate $\frac1{1+2}+\frac1{1+2+3}+\cdots+\frac1{1+2+\cdots+100}$.

In trapezium $ABCD$, $\angle BAD=\angle ADC=90^\circ$, $AC\perp BD$, $AB=\sqrt7$, $BC=\sqrt{217}$. Find $AD$.

Two numbers are independently and uniformly chosen from $[0,1]$. What is the probability that their absolute difference exceeds $1/4$?

Find the minimum of $\sum_{k=1}^{20}|n-k|$ over positive integers $n$.

A jury of $12$ needs an absolute majority to convict. Four will vote YES, three NO; four others independently toss fair coins for their votes. The last juror votes with the majority of the preceding eleven. Find the probability of conviction.

The array is shown. Define $S_i$ as the sum of entries in row $i$ or column $i$ of its top-left $i\times i$ square, counting the corner once. Given $S_1=1,S_2=8,S_3=27$, find $S_{21}$.

Find $x$ if $2^{x+1}+2^x+2^{x-1}=56\sqrt2$.

What is the maximum number of regions formed by drawing $100$ triangles in a plane?

In $\triangle ABC$, $\angle C=3\angle A$, $BC=3$, $AB=5$. Find $AC$.

Twelve students numbered $1$ to $12$ independently toss fair coins. Student $i$ scores $i$ for heads and $0$ for tails. Find the probability that the total score is divisible by $3$.

Acute angles $\alpha,\beta$ satisfy $\sin\alpha=2021\cos(\alpha+\beta)\sin\beta$. Find the maximum of $\tan\alpha$.

Place the integers $1$ to $9$ uniformly at random without repetition in a $3\times3$ table. Find the probability that every row sum and every column sum is odd.