Question bank SEAMO 2021

Real numbers satisfy $\dfrac{x+26}{y}+\dfrac{170}{xy}=\dfrac{2-y}{x}$. Evaluate $(x+y)^2$.

A recruit starts at $0$ on a number line and makes jumps, each either forwards or backwards, of lengths $2,4,8,16,32,64,128$ in that order. How many distinct final locations are possible?

How many four-digit integers have digits in strictly decreasing order from left to right?

How many ordered triples of positive integers $(a,b,c)$ satisfy $ab+bc+ca=abc+2$?

A family of $20$ parallel lines is intersected by a family of $21$ parallel lines. How many parallelograms are formed?

A clock has minute-hand length $4$ and hour-hand length $3$. From noon to $1$ pm inclusive, on how many occasions is the distance between their tips an integer?

How many ways are there to place $3$ indistinguishable rooks on a $5\times5$ chessboard so that no two share a row or column?

A circle with centre $O$ is tangent to a side of a square and passes through two vertices, as shown. The square has side $48$. Find the length of the broken path $AOBC$.

Positive real numbers satisfy $\log_3(\log_2x)+\log_{1/3}(\log_{1/2}y)=1$ and $xy=9$. Find $x+y$.

$2021$ students sit in a circle. Starting at student $A$, they count $1,2,3,1,2,3,\ldots$ clockwise; anyone counting $2$ or $3$ leaves immediately, until one remains. Find the survivor’s original position $n$, counting $A$ as position $1$.

Find the area enclosed by $x^2+y^2=|x|+|y|$ in the coordinate plane.

Among the integers in $A=\{1,2,\ldots,20\}$, let $m,n,p$ count those with exactly $2,3,4$ positive divisors respectively, including $1$ and the integer itself. Find $m+n+p$.

$ABC$ is equilateral and $P$ lies inside it with $PA^2=PB^2+PC^2$. Find $\angle BPC$ in degrees.

Find the last four digits of $1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+\cdots+99\cdot100\cdot3$.

Let $S=\{1,2,\ldots,21\}$. How many three-element subsets have element sum divisible by $3$?

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 $\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}=7$.