TIMO - 2020 - Grade 8 - Heat Round - Paper 2

If $\overline{abcd}+\overline{bcda}+\overline{cdab}+\overline{dabc}=17776$, calculate $a+b+c+d$.

There are 22 problems in a mathematics competition. The scores of each problem are allocated in the following ways: 3 marks will be given for a correct answer, 1 mark will be deducted from a wrong answer and 0 marks will be given for a blank answer. Find the minimum number of candidate(s) to ensure that 2 candidates will have the same scores in the competition.

30th September, 2020 will be Wednesday. Which day of the week will 28th September, 2028 be?

Given $A:B=9:11$, $B:C=6:12$, $C:D=7:13$ and $A+B+C+D=5800$, find $B$.

From the pattern $8,24,48,80,120,168,?,\ldots$, find the number replacing the question mark.

Given $a>2$ and that $ax^2+14x+6c=0$ has real roots, find the maximum integer value of $c$.

Factorize $x^2+8x-y^2-4y+12$.

How many integer solutions $x$ satisfy $-11\leq\frac{7x+12}{2}\leq29$?

Find $x$ in the system $\begin{cases}3x+y=30,\\5x-3y=8.\end{cases}$

Given $f(x)=ax^2+bx+c$, $f(-4)=3$, $f(4)=12$ and $f(-1)=1$, find $f(1)$.

A 3-digit odd number $x$ satisfies $x\equiv12\pmod{17}$. Find its maximum value.

Given $x>0$ and $x+\frac{1}{x}=7$, find $x^3+\frac{1}{x^3}$.

Find the last digit of $A=7+37+217+\cdots+(6^{99}+1)+(6^{100}+1)$.

If $(x-1)$ is a factor of $2019x^2-1994x-c$, find $c$.

Find the sum of positive factors of 836.

How many diagonals does a convex dodecagon (12-sided polygon) have?

Given right-angled triangles $\triangle ABC\sim\triangle DEF$ with legs $AB=15$ and $AC=36$, find $\frac{DE}{EF}$.

Line $L$ passes through $A(5,-1)$ and has slope 10. Find its $y$-intercept.

Given points $A(3,-7)$, $B(-8,-1)$ and $C(9,4)$ on a coordinate plane, find the area of triangle $ABC$.

An interior angle of a regular $n$-sided polygon is $(12n+48)^\circ$. Find the maximum value of $n$.