TIMO 2020 - Senior Secondary (Grades 10-12) - Preliminary Round - Paper 2

Two trains are 85 m and 65 m long. They completely pass each other in 6 seconds when travelling in opposite directions, and in 30 seconds when travelling in the same direction. Find the faster train's speed in km/h.

During Black Friday week, a store's daily sales increase by 20% over the previous day. By what percentage have sales increased after two days compared with the original sales?

How many values of $x$ make both $x+2$ and $3x+4$ two-digit primes?

Each letter in the illustrated addition represents a different digit. Find $I$.

An archer shoots 30 arrows. An inner-ring hit earns 10 points, an outer-ring hit earns 5, and a miss loses 5. His final score is 100. What is the minimum number of misses?

Find $z$ if $\lvert2020-2x+y\rvert+(x-y-2021)^2+\sqrt{x+y+z}=0$.

Find the minimum value of $\frac{4a^2+9}{a+2}$ for a positive real number $a$.

How many positive integers $a$ satisfy $a+2<\frac{2a+20}{a-2}$?

Find the coefficient of $x^6$ in $(2x^2-1)^{10}$.

$2x^3+ax^2+bx+3$ is divisible by $2x^2+x-1$. Find $a+b$.

Find the largest integer $n$ for which $\frac{123!}{6^n}$ is an integer.

A three-digit number leaves remainder 4 when divided by 5 and remainder 1 when divided by 4. How many such numbers are there?

Find the remainder when $98^{100}$ is divided by 20.

Find the units digit of $2+2^2+2^3+\cdots+2^{2020}+2^{2021}$.

Linda concatenates consecutive natural numbers to form $A=1234567891011121314\ldots$. What is the minimum number of digits $A$ can have if it is divisible by 45?

A circle has centre $O(3,4)$ and diameter $MN$, where $M(2+x,y)$ and $N(-2-y,x)$. Find $4x+2y$.

Susan draws a regular $n$-gon, then erases two vertices to form an $(n-2)$-gon. The ratio of the sums of their interior angles is $10:9$. Find $n$.

A right triangle has legs 6 and 8. Its incircle and circumcircle have diameters $d$ and $D$. Find $d+D$.

Triangle $ABC$ is right-angled at $A$, with $AB=30$, $AC=40$. $M$ is the midpoint of $BC$ and $AH$ is the altitude. Find $MH$.

Figure 1 consists of three identical rectangles with the dimensions shown. Cutting along the dotted line gives regions $A,B,C$ in Figure 2. Find the ratio of the area of $A$ to that of $C$.