TIMO - 2021 - Grade 7 - Heat Round - Paper 5

There are some chickens and rabbits in a cage. The number of rabbits is 4 times as the number of chickens. The total number of rabbit’s legs is 126 more than that of chicken. How many rabbit(s) is / are there?

Alice goes north for 18km, then goes east for 24km. How many km is she now from the original position?

Peter bought fruit from a market. It is known that a watermelon, an apple and an orange cost 32 dollars, 4 dollars and 2 dollars respectively. He spent 440 dollars to buy 20 fruits and bought at least 1 for each type. How many apples did he buy?

According to the pattern $1,7,19,43,91,?,\ldots$, find the number replacing the question mark.

There are 7 people arguing which day of the week is today. A: Yesterday was Monday. B: Tomorrow will be Friday. C: Tomorrow will be Thursday. D: The day after tomorrow will be Wednesday. E: Today is neither Monday, Wednesday nor Thursday. F: Today is neither Tuesday, Thursday nor Saturday. G: Today is either Wednesday, Thursday or Friday. Given that among them only one statement is correct, which day of the week is today?

Find $x$ if $(6x-2)-(2x+6)=16$.

How many integer solutions $x$ satisfy $-3<\frac{3x+5}{2}<19$?

Given integer $a$ with $\lvert a-2\rvert+\lvert a+4\rvert=6$, find the sum of all possible values of $a$.

Factorize $3x^2-9x-210$.

$a$ is a negative real number and $a
e-1$. If $a+\frac{18}{a+1}=-10$, find the minimum value of $a$.

When a 3-digit number is divided by 4, 6 or 9, the remainder is 3. What is the maximum possible value of that 3-digit number?

Find the units digit of $A=1^1\times9^2\times11^3\times19^4\times21^5\times\cdots\times91^{19}\times99^{20}$.

Find the remainder when $27^{38}$ is divided by 13.

The difference between two square numbers is 115. Find the minimum value of the larger square.

How many simplified fractions with denominator 1215 are there? Here, as stated in the source glossary, simplified fractions mean positive fractions less than 1.

$M(3y,2x)$ is the midpoint of $A(x,8)$ and $B(21,y)$. Find $x+y$.

An interior angle of a regular n-sided polygon is 8 times of an exterior angle. Find the value of n.

Combine 678 equilateral triangles with side length 1 unit to form a quadrilateral. Find the minimum perimeter of that quadrilateral.

A circle has radius 14, and its area is 14 times the perimeter of a square. Find the area of the square. Take $\pi=\frac{22}{7}$.

The figure below is overlapped by a big equilateral triangle with side length 1 and a smaller square. Find the area of the square. (Answer in surd form)