SEAMO 2027 - Paper F - SET 10 - 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.

Find $a+b$ if $\frac{x^2}{(x^2+1)(x^2+4)}=\frac a{x^2+1}+\frac b{x^2+4}$.

Interpret the displayed relation as an identity in the real variable x, with a and b constant.

Andrew and Bob play a game with five cards numbered 1, 2, 3, 4 and 5. Andrew shuffles them and places them face-down on a table. Each round Bob chooses any three cards and Andrew states the value of one of those three. Bob may repeat this as often as he likes. How many cards at most does Bob need to choose to ensure that he knows the positions of all the cards?

“Ensure” means a guarantee for every permitted truthful choice of reply by Andrew; no cooperative or random reply rule is assumed.

How many positive integers less than 2020 have digit sum 9?

A class has 5 boys and 6 girls. Each student receives 1, 2 or 5 dollars. No two students of opposite gender receive the same amount. How many possibilities are there?

Counting convention: all students are distinct. The dollar currency is not further identified by the source.

Let $m,n$ be real numbers with $m$ nonzero. Suppose $a,b,c$ are the roots of $mx^3-mx^2+nx+n=0$. Find $(a+b+c)(1/a+1/b+1/c)$.

Domain clarification for self-study: assume n is also nonzero, so every reciprocal in the requested expression is defined. The original source states only that m is nonzero.

Let $\alpha,\beta$ be the roots of $x^2+x-3=0$. Which choice is the value of $\alpha^3-4\beta^2+20$?

If $g(x)=\tan(x/2)$ for $0\lt x\lt \pi$ and $f(g(x))=\sin 2x$, find k such that $k f(\sqrt2/2)=36\sqrt2$.

In quadrilateral $ABCD$, $\angle ACB=\angle BAD=105^\circ$ and $\angle ABC=\angle ADC=45^\circ$. Which statement is true?

Assign the numbers 4, 5, 6, …, 11 randomly, one to each vertex of a cube. What is the probability that the numbers on every pair of adjacent vertices are relatively prime?

Suppose $x,y$ are real numbers such that $x-y=8$ and $x^2+y^2=194$. Find $xy$.

Find the number of representations
$$2072=a\cdot10^3+b\cdot10^2+c\cdot10+d,$$
where $a,b,c,d$ are integers with $0\le a,b,c,d\le99$.

Positive real numbers $a,b,c$ satisfy $ab+bc+ca+a^2=45$, $ab+bc+ca+b^2=50$, and $ab+bc+ca+c^2=90$. Evaluate $a+b+c$.

It is known that $f(x_n)=\frac{n+1}{2n+1}$, $n\in\mathbb N$. Then $f(x_n)$ is:

The original choices overlap and one term has an ambiguous convention. Give any one correct choice or a correct description of the sequence. This is assessed manually; no extra proof or complete list of choices is required.

Original choices:

A. strictly decreasing

B. non-strictly decreasing

C. monotonic decreasing

D. strictly increasing

E. None of the above

Find the least positive integer n such that $\lfloor10^n/x\rfloor=98$ has an integer solution x. Here $\lfloor k\rfloor$ is the greatest integer less than or equal to k.

In the expansion of $f(x)=(1+ax)^4(1+bx)^5$, where $a,b$ are positive integers, the coefficient of $x^2$ is 66. Evaluate $a+b$.

Source-consistency warning: the printed numerical data and supplied answer key are inconsistent. Retain the expression and all data exactly as printed, and assess the listed choices from those data.

Find the remainder when $3^{2021}$ is divided by 31.

Let $a_n=\underbrace{111\cdots1}_{n\text{ ones}}$ for each positive integer $n$. Find the least $n$ for which $a_n$ is divisible by 19.

Given that a positive integer is a four-digit palindrome, what is the probability that it is a multiple of 44?

Choose uniformly from all four-digit palindromes. A discrepancy with the supplied key is explained in the solution; answer from the stated data.

Evaluate $\frac12+\frac{2^2}{2^2}+\frac{3^2}{2^3}+\frac{4^2}{2^4}+\cdots+\frac{n^2}{2^n}+\cdots$.

Find the number of ordered triples of positive integers $(a,b,c)$ with $1\le a,b,c\le10$ such that
$$a^2+b^2+c^2+2ab+2a(c-1)+2b(c+1)$$
is a perfect square.