SEAMO 2027 - Paper E - SET 07 - Self-study

SEAMO self-study edition 2027; not an official 2027 paper. Original source years are retained. 25 questions, 90 minutes; Q1–10: 3 points, Q11–20: 4 points, Q21–25: 6 points, total 100. Original response types and all subparts are preserved. Verified is an editorial convention; source defects, necessary assumptions and unresolved mathematical uncertainty are disclosed.

Suppose $a,b,c,d,e$ are consecutive positive integers with $a\lt b\lt c\lt d\lt e$. The sum $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube. Find the smallest value of $c$.

Enter the exact number only; fractions are allowed.

$ABCD$ is a rectangle. Points $X$ and $Y$ lie on $BC$ and $CD$, respectively, and the areas of triangles $ABX,CXY,AYD$ are $3,4,5$, respectively. Find the area of triangle $AXY$.

Find $a$ such that $\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}=7$.

The infinite digit sequence $123456789101112131415\ldots$ is obtained by writing the positive integers in order. What is its $2022$nd digit?

Simplify $$\sqrt{\log_2 3\,\log_2 12\,\log_2 48\,\log_2 192+16}-\log_2 12\,\log_2 48+10.$$

How many arrangements of $A,A,B,B,C,C,D,E$ in a straight line have no identical letters next to each other?

Enter the exact number only; fractions are allowed.

Let $m=\overline{2x192y20}$ be an eight-digit integer. If $m$ is divisible by $41$, how many digit pairs $(x,y)$ are possible?

Goldbach’s conjecture suggests that every even integer greater than $2$ is a sum of two primes. The number $186$ can be so expressed. What is the greatest possible product of the two primes?

Beatrice has some nuggets. Counting them in groups of $5,3,11$ leaves remainders $2,2,3$, respectively. What is the smallest possible number of nuggets?

Enter the exact number only; fractions are allowed.

Given $$\frac{ab}{a+b}=2,\qquad\frac{ac}{a+c}=5,\qquad\frac{bc}{b+c}=4,$$ find $a+b+c$.

Enter the exact number only; fractions are allowed.

Find the last two digits of $$\underbrace{18^{2018}+18^{2018}+\cdots+18^{2018}}_{8\times18^{2018}\text{ terms}}.$$

Neha added the page numbers $1,2,3,\ldots,n$ of a book and obtained $2018$. She then realised that one page number had been added twice. Find that number.

Which expression equals $\log\frac{25}{16}-2\log\frac59+\log\frac{32}{81}$?

The integers $1$ to $9$ are placed row by row in the $3\times3$ array shown. Its diagonal sum is $15$. If a similar array is constructed with the integers $1$ to $100$, what is its diagonal sum?

Find $x$ in $$\log\frac{13}{19}+\log\frac{95}{91}-\log\frac57+\log5=\log x.$$

In a rectangular coordinate system, two circles pass through $(3,2)$ and are tangent to both coordinate axes. Find the sum of their radii.

Enter the exact number only; fractions are allowed.

Find the remainder when $$2020!\left(1+\frac12+\frac13+\cdots+\frac1{2020}\right)$$ is divided by $2021$.

Alloys A, B and C contain the following percentages of lead, tin and zinc, respectively: A: $(30,70,0)$; B: $(0,80,20)$; C: $(10,50,40)$. Terry mixes these alloys to obtain alloy T containing $15\%$ lead. If $m,n$ are the maximum and minimum possible tin percentages, find $m-n$.

Allow a zero amount of one starting alloy; percentages refer to mass and mixing preserves mass.

Enter the exact number only; fractions are allowed.

Two fair dice are thrown. Consider these claims about the difference between the numbers: (i) the probability of $2$ is $2/9$; (ii) of $3$ is $1/6$; (iii) of at least $2$ is $1/3$; (iv) of at most $4$ is $7/9$. Which claims are true?

Difference means the nonnegative absolute difference; the two fair dice are independent.

$ABCD$ is a square of side $10\text{ cm}$. Points $E,F$ vary on $BC,CD$, respectively, with $BE=CF$. Find the smallest area of triangle $AEF$, in $\text{cm}^2$.