SEAMO X 2019 — Paper E — Grade 9–10

Find the remainder when $2019^1+2019^2+\cdots+2019^{2019}$ is divided by $101$.

Find the minimum of $2019-\dfrac{18}{x^2-4x+5}$ for real $x$.

Rectangle $ABCD$ is inscribed in a circle, and $E$ lies on the circumference. Given $AE^2+BE^2+CE^2+DE^2=200$ and the circle’s area is $k\pi$, find $k$.

How many ordered triples of nonnegative integers $(a,b,c)$ satisfy $abc+ab+bc+ca+a+b+c=2019$?

Evaluate $N=\displaystyle\prod_{j=1}^{1009}\frac{\frac1{2j}-\frac1{2j+1}}{\frac1{2j+1}-\frac1{2j+2}}$. The first factors use $2,3,4$ and $4,5,6$; the last uses $2018,2019,2020$.

Evaluate $\dfrac{\tan50^\circ\tan60^\circ\tan70^\circ}{\tan50^\circ+\tan60^\circ+\tan70^\circ}$.

Nicholas chooses five different numbers from $41,47,48,56,64,67,78$. Their average is an integer. If $47$ and $64$ are chosen, find the sum of the other three.

Two chess teams each have $7$ members. One game is played at a time, with one representative per team. The winner stays, the loser is eliminated, and eliminated players never return. How many ways can the competition progress?

A $2019$-digit integer is $\overline{a_1a_2\ldots a_{2019}}$. For every $i=1,\ldots,2018$, the two-digit number $\overline{a_i a_{i+1}}$ consists of $3$ distinct prime factors. Find $a_{2019}$.

In the figure $AB\perp BC$, $D\in AB$, $E\in BC$, and $BA+AE=BD+DC$. Given $AD=2$, $BE=3$, $EC=4$, find $BA+AE$.

Positive real numbers satisfy $a\sqrt b=a+c$, $b\sqrt c=b+a$, $c\sqrt a=c+b$. Evaluate $a+2b+3c$.

Integers $a,b$ make $\sqrt{4-2\sqrt3}$ a root of $x^2+ax+b=0$. Find $a+b$.

In $(\sqrt x+2)^{2n+1}$, find the sum of coefficients of the terms whose power of $x$ is an integer. Give the answer in terms of $n$.

For real numbers $a\gt 1,b\gt 1$, find the minimum of $\dfrac{\log_a2019+\log_b2019}{\log_{ab}2019}$.

In triangle $ABC$, $\angle A=\alpha$, and $CD,BE$ are the altitudes to the lines $AB,AC$ respectively. Find $DE/BC$ in terms of $\alpha$.