TIMO - 2019 - Grade 9 - Heat Round - Paper 3

The sum of 11 consecutive odd numbers is 253. Find the third largest number.

There are 25 problems in a mathematics competition. The scores of each problem are allocated in the following ways: 4 marks will be given for a correct answer and 0 marks will be given from a wrong answer or for a blank answer. Find the minimum number of candidate(s) to ensure that 5 candidates will have the same scores in the competition.

If $\overline{abcd}+\overline{bcda}+\overline{cdab}+\overline{dabc}=26664$, find $a+b+c+d$.

Given $A:B=3:5$, $B:C=7:11$, $C:D=2:3$ and $A+B+C+D=2322$, find $C$.

13th October 2018 is Saturday. Which day of the week will 13th October 2024 be?

If $\alpha,\beta$ are the roots of $x^2-9x+13=0$, find $\alpha\beta^2+\alpha^2\beta$.

If $x^2+8y+y^2-6x+25=0$, find $x-y$.

Simplify $\sqrt{28+10\sqrt3}$.

Find $t$ in $\begin{cases}r+s+t+u+4v=18\\r+s+t+4u+v=19\\r+s+4t+u+v=20\\4r+s+t+u+v=21\\r+4s+t+u+v=22\end{cases}$.

Rational $x>0$ satisfies $x=\sqrt{5\sqrt{5\sqrt{5\sqrt{\cdots}}}}$. Find $x$.

A 3-digit number has exactly 9 positive factors. Find the maximum value of this 3-digit number.

Let $x>0$ be a 3-digit odd number with $x\equiv19\pmod{23}$. Find the minimum value of $x$.

How many reduced positive fractions less than 1 have denominator 1013?

A 4-digit number has a remainder of 6 when divided 7, has a remainder of 7 when divided by 8, has a remainder of 8 when divided by 9. How many such possible four- digit numbers are there?

Rational $x>0$ satisfies $x=2+\frac5{4+\frac5{4+\cdots}}$. Find $x$.

Find the area enclosed by the x-axis, y-axis and the line $4x-y+16=0$.

In quadrilateral $ABCD$, $AC\perp BD$ and the diagonals intersect at $P$. Given $AB=13$, $BC=7$, $CD=1$, find $DA$.

Line $L$ passes through $A(-5,7)$ and has slope 4. Find its y-intercept.

Find the area of the triangle with vertices $A(4,3)$, $B(7,12)$, $C(3,2)$ in a rectangular coordinate system.

In triangle $ABC$, $AB=36$, $BC=18$, $AC=19$. Find $\frac{\sin A}{\sin C}$.