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SEAMO 2026 Paper F SET 2 – Self-practice

Question 1

Find the last two digits of $11^{2020}$.
  1. 01
  2. 41
  3. 71
  4. 91
  5. None of the above

Question 2

The quadratic equation $x^2-52x+k=0$ has roots that are prime numbers. Find the maximum value of k.
  1. 520
  2. 576
  3. 640
  4. 667
  5. None of the above

Question 3

Let $f(x)=x^2+2020x+20$. How many ordered pairs of positive integers (m,n) satisfy $f(m+n)=f(m)+f(n)$?
  1. 1
  2. 2
  3. 3
  4. 4
  5. None of the above

Question 4

Find the sum of all positive integers n for which $\dfrac{4n^3-16n^2+29n+60}{2n-3}$ is an integer.
  1. 42
  2. 69
  3. 75
  4. 81
  5. None of the above

Question 5

Evaluate $S=\dfrac1{1+1^2+1^4}+\dfrac2{1+2^2+2^4}+\dfrac3{1+3^2+3^4}+\cdots+\dfrac{20}{1+20^2+20^4}$.
  1. 1/2
  2. 210/421
  3. 105/211
  4. 1
  5. None of the above

Question 6

2 white, 3 black and 4 grey marbles are shared equally among 9 students. Find the number of distributions such that Bran and Sansa get the same colour and Arya gets a grey marble.
  1. 120
  2. 130
  3. 140
  4. 150
  5. None of the above

Question 7

Given that a is a real number such that $a^4+a^3+a^2+1=0$, evaluate $a^{2020}+2a^{2010}+3a^{2000}$.
  1. 2
  2. 4
  3. 6
  4. 8
  5. None of the above

Question 8

Given distinct real numbers a,b,c such that $a+\frac1b=b+\frac1c=c+\frac1a$, what is the largest possible value of abc?
  1. 1/2
  2. 2
  3. 5/2
  4. 3
  5. None of the above

Question 9

How many positive integers less than 2020 have digit sum 9?
  1. 50
  2. 100
  3. 102
  4. 202
  5. None of the above

Question 10

The sequence $\{a_n\}$ is defined by $a_{n+2}=\dfrac{1+a_{n+1}}{a_n}$, with $a_1=1,a_2=2$. Evaluate $a_{2020}$.
  1. 1
  2. 2
  3. 3
  4. 4
  5. None of the above

Question 11

Find the smallest prime factor of $1\underbrace{00\cdots0}_{2020\text{ zeros}}1$.
  1. 3
  2. 5
  3. 7
  4. 11
  5. None of the above

Question 12

In the expansion of $f(x)=(1+ax)^4(1+bx)^5$, where a and b are positive integers, the coefficient of $x^2$ is 66. Evaluate a+b.
  1. 2
  2. 3
  3. 4
  4. 5
  5. None of the above

Question 13

The equation $x^3-ax^2+bx-2020=0$ has three positive integer roots. Find the least possible value of a.
  1. 101
  2. 110
  3. 202
  4. 220
  5. None of the above

Question 14

Evaluate $S=\sin^2 0^\circ+\sin^2 2^\circ+\sin^2 4^\circ+\cdots+\sin^2 180^\circ$.
  1. 80
  2. 81
  3. 88
  4. 90
  5. None of the above

Question 15

Given real numbers a,b satisfying $6-5a+4b-3a^2+2ab-b^2=0$ and $a-b=1$, find the sum of all possible values of $30a/b$.
  1. -15
  2. -10
  3. 15
  4. 30
  5. None of the above

Question 16

The figure shows a 5 × 6 rectangular board with a missing 1 × 2 rectangle in the centre. How many squares are there in the board?

Question illustration
  1. 14
  2. 30
  3. 54
  4. 56
  5. None of the above

Question 17

In triangle ABC, $(\sin A+\sin B):(\sin B+\sin C):(\sin C+\sin A)=19:20:21$. Find $99\cos A$.
  1. 39
  2. 41
  3. 51
  4. 60
  5. None of the above

Question 18

Find the least positive integer n such that $\left\lfloor\dfrac{10^n}{x}\right\rfloor=98$ has an integer solution x. Here $\lfloor k\rfloor$ is the largest integer less than or equal to k.
  1. 3
  2. 4
  3. 5
  4. 6
  5. None of the above

Question 19

How many positive integers k<100 make $2(5^{6n})+k(2^{3n+2})-1$ divisible by 7 for every positive integer n?
  1. 12
  2. 14
  3. 18
  4. 19
  5. None of the above

Question 20

A,B,C,D are four distinct points on a circle. Chords AB and CD are perpendicular at E. Given EA=4, EB=24 and EC=6, find the radius of the circle.

Question illustration
  1. $\sqrt{221}$
  2. 15
  3. $\sqrt{270}$
  4. 18
  5. None of the above