SEAMO X 2020 — Paper E — Grade 9–10

$1572,2448,3762$ leave the same remainder when divided by a two-digit positive integer $n$. Find $n$.

How many positive integers $n$ make $3465+n^2$ a perfect square?

Three semicircles have the sides of a right triangle as their respective diameters. Each labelled region $A,B,C,D,E$ has integer area and the total area is $12$. If $A+B$ is minimum, find $E$.

Given $x=\dfrac{13}{\sqrt{19+8\sqrt3}}$, find the exact value of $A=\dfrac{x^4-6x^3-2x^2+18x+23}{x^2-8x+15}$.

The medians of a triangle have lengths $15,15,18$. Write its perimeter as $a+b\sqrt c$ with integer $a,b,c$ and $c$ minimal in the usual square-free radical form. Find $a+b+c$.

Positive integers satisfy $ab+bc=21$, $bc+ca=25$, $ca+ab=16$. Find $30\left(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}\right)$.

Let $S_n=\dfrac{n^{99}+99^n}{10}$. If $x$ is the fractional part of $S_{2020}$, find $10x$.

A fair six-sided die is thrown until the first $1$ appears. The probability that an odd number of throws is needed is $p/q$ in lowest terms. Find $p+q$.

For $\lfloor x\rfloor$ denoting the greatest integer not exceeding $x$, evaluate $S=\left\lfloor\frac{2^2}3\right\rfloor+\left\lfloor\frac{3^2}4\right\rfloor+\cdots+\left\lfloor\frac{99^2}{100}\right\rfloor$.

Triangle $ABC$ is right-angled at $C$. The bisector $BD$ of $\angle ABC$ meets $AC$ at $D$. Given $[ABD]=36$ cm² and $CD=12$ cm, find $AB$ in cm.

Positive real numbers satisfy $a+\lfloor b\rfloor+\{c\}=11.1$, $\lfloor a\rfloor+\{b\}+c=12.2$, and $\{a\}+b+\lfloor c\rfloor=13.3$, where $\{x\}=x-\lfloor x\rfloor$. Find $\lfloor a\rfloor+\lfloor b\rfloor+\lfloor c\rfloor$.

Find the integer part of $A=\dfrac1{\sqrt2}+\dfrac1{\sqrt3}+\cdots+\dfrac1{\sqrt{100}}$.

Real numbers $x_1,\ldots,x_{100}$ satisfy $x_1^2+2x_2^2+\cdots+100x_{100}^2=1$. Find the greatest integer smaller than or equal to $x_1+2x_2+\cdots+100x_{100}$.

For how many positive integers $n$ is $(n^3-1)/5$ prime?

For real $x$, let $f(x)=(19-x)(21-x)(19+x)(21+x)$. If $k$ is its minimum, find $|k|$.