SEAMO 2026 Paper F SET 1 – Self-practice

Find the smallest 4-digit number that has the same number of factors as 2022.

In triangle ABC, $\angle C=90^\circ$, CA=3 and CB=4. D is on segment AB with CD perpendicular to AB. E is on segment AC with DE perpendicular to CA. If $DE=m^2/n^2$, where m and n are relatively prime, find m+n.

If x and y are real numbers satisfying $x^2-2xy+2y^2-4y+4=0$, find $x^4+y^4$.

Let O and I be the circumcentre and incentre of triangle ABC. Its sides are 6,8,10 and $OI=x$. Find $x^2$.

Find the last 2 digits of $2022^{2022}$.

Andrew and Bob play with 2022 face-down cards:1001 black and the rest white. Each round, two cards are drawn consecutively. Black then white means Andrew wins; white then black means Bob wins. If the colours agree, continue without replacement. If Andrew’s winning probability is x, find $\lfloor2022x\rfloor$.

Let $\alpha=\pi/18$. Find $\cos\alpha+\cos2\alpha+\cdots+\cos18\alpha$.

Find the remainder when $20^{2027}+2^{2027}+23^{2027}+32^{2027}$ is divided by 2027.

Let $f(x)=(x+1)^2$. Find the sum of all values of x that fulfil $fff(x)=x$.

In triangle ABC, $AC=2\sqrt{17}$, BC=8 and AB=10. D,E,F are the midpoints of BC,CA,AB respectively. P,Q are the projections of A,D on EF. Find PQ.

Find the number of ways to represent 2072 as $2072=a\cdot10^3+b\cdot10^2+c\cdot10+d$, where a,b,c,d are whole numbers and $0\le a,b,c,d\le99$.

ABCD is a quadrilateral with AB=BC=CD=DA and $\angle ABC<90^\circ$. Points E,F are on lines BC,CD respectively, and AB=AE=AF=FE. Find $\angle BEF$.

Any collection of n distinct integers between 1 and 20 contains at least one pair such that one number is a multiple of the other. Find the smallest n.

A three-digit number $\overline{abc}$ (a,b,c need not be distinct) has no zero digit. If $\overline{abc},\overline{bca},\overline{cab}$ are all divisible by 7, find the sum of all possible $\overline{abc}$.

In triangle ABC with sides a,b,c, $a^2+b^2+c^2$ is 20 times its area. Find $\cot A+\cot B+\cot C$.

The probability of rain is 3/5. Andrew and Bella go home from school. Andrew brings an umbrella with probability 1/3; Bella does so with probability b. If it does not rain, they go together with probability c. If it rains, they go together if and only if exactly one brings an umbrella. Their probability of going together is 2/5. The minimum of $b^2+c^2$ is δ. Find $\lfloor100\delta\rfloor$.

Let $f(x)=\sqrt{\log_2\frac{5x+1}{4x-1}}$. If the function is not defined for $m<x\le n$, find $\lfloor m+100n\rfloor$.

An arrangement of the digits 0,1,2,…,9 is called cheerful when 0,1,2,3,4 occur in ascending order and 5,6,7,8,9 occur in descending order. For example 9801237465 is cheerful. If the first digit cannot be 0, how many cheerful arrangements are there?

The polynomial $1-x+x^2-x^3+\cdots+x^8-x^9$ can be written as $a_0y^9+a_1y^8+a_2y^7+\cdots+a_8y+a_9$, where x=y−1. Find $|a_2|$.

Andrew and Bob play with 5 cards numbered 1,2,3,4,5. Andrew shuffles and places them face-down. Each round Bob chooses any 3 cards, and Andrew states the value of one of those cards. Bob may ask as many rounds as he likes. How many cards at most does he need to choose to ensure he knows the positions of all the cards?