SEAMO 2027 - Paper E - SET 02 - 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.

Let $x,y$ be real numbers with $x>1$ and $y>1$. Find the smallest value of $$\frac{\log_x2011+\log_y2011}{\log_{xy}2011}.$$

Points $P$ and $Q$ are the centres of the circles shown. Chord $AB$ is tangent to the circle with centre $P$. Given $PQ\parallel AB$ and $AB=24$ cm, find the area of the shaded region.

The source does not specify a numerical approximation for π. For this practice question, compare the options using exact circle areas; the source-key convention is discussed after answering.

How many real solutions $(x,y,z)$ does the following system have? $$x+y=2,\qquad xy-z^2=1.$$

Let $x,y,z$ be real numbers less than $4$. Prove that among $$\frac1x+\frac1{4-y},\qquad\frac1y+\frac1{4-z},\qquad\frac1z+\frac1{4-x}$$ there is at least one number greater than or equal to $1$.

The printed assumptions contain a domain/validity defect. Retain those assumptions and assess whether the requested claim follows; identify undefined cases or counterexamples if necessary. Any additional condition must be stated as a correction, not as an original given.

If $a+b=2$ and $a^2+b^2=5$, find $a^3+b^3$.

Prove that for every odd natural number $n$, the number $$1^{2007}+2^{2007}+\cdots+n^{2007}$$ is not divisible by $n+2$.

Evaluate $$\frac{300^2}{253^2-247^2}.$$

The sum of the first $n$ terms of $\{a_n\}$ is $S_n=2a_n-1$, for $n=1,2,3,\ldots$. The sequence $\{b_n\}$ satisfies $b_1=3$ and $b_{k+1}=b_k+a_k$, for $k=1,2,3,\ldots$. Find the sum of the first $n$ terms of $\{b_n\}$ when $n=8$.

Find the remainder when $2005^{\left(2007^{2009}\right)}$ is divided by $7$.

$A,B,C,D$ lie on a circle. $AC$ is a diameter and $AD=DC$. The area of $ABCD$ is $20$ cm$^2$. Find $DE$, where the diagram marks $DE\perp AB$.

In trapezium $ABCD$, $AD\parallel BC$, $AB\perp BC$, $AB=BC$ and $AE=AD$. Point $E$ lies on $AB$ and $\angle BEC=75^\circ$. Find $\angle DCE$.

Evaluate $$\frac{\cos^4 15^\circ+\sin^4 15^\circ+2\sin^2 15^\circ\cos^2 15^\circ}{\cos^6 15^\circ+\sin^6 15^\circ+3\sin^2 15^\circ\cos^2 15^\circ}.$$

$n$ is a positive integer such that $n^2+5n+13$ is a perfect square. Find $n$.

Two numbers are selected at random from $1,2,3,\ldots,100$ and multiplied together. Find the probability that the product is divisible by $3$.

Practice convention: select two distinct numbers uniformly, without replacement. The source does not explicitly say whether replacement is allowed.

Find the largest positive integer $n$ such that $3^{1024}-1$ is divisible by $2^n$.

The distance from Georgetown to Greenville is $120$ km. An SUV and a sedan leave Georgetown for Greenville at the same time. After one hour, the SUV is $24$ km ahead of the sedan. The sedan arrives $25$ minutes after the SUV. Find the speed of the sedan.

Use constant speeds throughout the journey, as intended in this travel-time problem.

A square $ABCD$ and an equilateral triangle $PQR$ are inscribed in a circle with centre $O$, with $AB\parallel QR$. Sides $PQ$ and $PR$ meet $AB$ at $X$ and $Y$, respectively. Find $\tan\angle XOY$.

The original figure swaps X and Y relative to the textual order of the intersections. Both the original text and figure are retained; use the text to name the points. Exchanging X and Y does not change the undirected angle XOY.

Evaluate $$\frac{2020}{\sqrt1+\sqrt5}+\frac{2020}{\sqrt5+\sqrt9}+\frac{2020}{\sqrt9+\sqrt{13}}+\cdots+\frac{2020}{\sqrt{2021}+\sqrt{2025}}.$$

Find the remainder when $6^{273}+8^{273}$ is divided by $49$.

Nine cards are labelled $1,2,2,3,3,3,4,5,6$. Cards with odd numbers are red and the others are blue. How many arrangements in a row have the number on each red card less than or equal to the number on every red card to its right?

Practice counting convention: cards with the same number and colour are indistinguishable; arrangements are distinguished by the visible row of numbers.

Enter the number only.