SMO 2025 Senior Round 1

Suppose the roots of $x^2+Ax+B=0$ are $A+B$ and $B-A$, where $A\ne B$. Find the value of $A-B$.

Suppose $2025\log_{2^{2025}}\sqrt{2025}+\dfrac{2}{\log_{\sqrt{2025}}2}=\log_2 M$. Find $M$.

Suppose $y=\sin^2x-4\sin x+15$, where $x$ is a real number. Find the maximum possible value of $y$.

Find the value of $\dfrac{\sin75^\circ+\sin15^\circ}{\cos75^\circ+\cos15^\circ}$.

Find the maximum possible value of $\dfrac{3x^2}{x^2+2x+2}$.

Find the maximum value of $16-64\sin x\cos x+48\sin^2x$.

Find the largest coefficient in the expansion of $\left(\dfrac1x+3x^2\right)^8$.

Suppose $x$ and $y$ are two real numbers such that $x-y=10$ and $x^2-2xy+y^2+xy^2-y^3=910$. Find the maximum possible value of $x+y$.

Suppose $x$ is a real number satisfying $\left(x-\dfrac53\right)^{5+\sqrt{x-11/3}}=\dfrac{96}{x-2/3}$. Find $9x^2$.

Let $f(x)=\dfrac{2^{x+2}}{2^x+1}$. Find $f(-1/2025)+f(-1/2024)+\cdots+f(-1)+f(0)+f(1)+\cdots+f(1/2024)+f(1/2025)$.

Let $m$ be the minimum value of $\dfrac{14-\sin^2 2\theta-4\cos^4\theta}{\sin2\theta}$ for $0\lt \theta\lt \pi/4$. Find $m^2$.

Let $g(x)=x^3+3x^2-x-5$ and suppose that $\alpha_1,\alpha_2,\alpha_3$ are the three distinct real roots of $g(x)$, where $\alpha_1\lt \alpha_2\lt \alpha_3$. Find $|\lfloor\alpha_1\rfloor|+|\lfloor\alpha_2\rfloor|+|\lfloor\alpha_3\rfloor|$.

Let
$$N=\sum_{k_1=0}^{2025}\sum_{k_2=0}^{k_1}\sum_{k_3=0}^{k_2}\cdots\sum_{k_{2098}=0}^{k_{2097}}
\frac{n!}{(n-k_1)!(k_1-k_2)!\cdots(k_{2097}-k_{2098})!\,k_{2098}!}.$$
Find the remainder when $N$ is divided by $3000$.

The supplied scan includes the handwritten annotations $n=2025$ and the terminal factor $k_{2098}!$; both are retained in the transcription above and shown in the original excerpt below.

A biased five-sided dice has faces labelled $1,2,3,4,5$ and has probability $p$ of coming up in an even number. Given that the probability that three tosses resulted in an odd number in six independent tosses of this dice is $8/225$, and that the possible values of $p$ are $p_1,p_2$, find $\left\lfloor\dfrac1{p_1p_2}\right\rfloor$.

Let $C_1$ be the curve with $y=x^2$. Suppose that $C_1$ is rotated anticlockwise by $75^\circ$ to obtain the curve $C_2$. It is known that the equation of $C_2$ can be written in the form $(8-4\sqrt3)x^2+By^2+Cx+Dy+Exy=0$. Find $\left\lfloor(8-4\sqrt3)+|B|+|C|+|D|+|E|\right\rfloor$.

Let
$$A_{m,n}=\frac1{\log_2 2025\,\log_3 2\,\log_4 3\cdots\log_{99}98\,\log_{2025^{mn}}99}$$
and let $S_k=A_{2025,2}+A_{2025,3}+\cdots+A_{2025,k}$. If $T=\sum_{k=2}^{\infty}1/S_k$, find $\lceil1/T\rceil$.

The lines $y=-\dfrac43x+\dfrac{40}3$ and $y=\dfrac43x+\dfrac{40}3$, and the $x$-axis are tangents to a circle of radius $r$. Find $r$.

It is given that $\mathcal E=\{1,2,3,4,5,6,7\}$, and $A\subseteq\mathcal E$, $B\subseteq\mathcal E$, $C\subseteq\mathcal E$ are such that $A\cap C=\varnothing$ and $(A\cap B)\cup(B\cap C)=\{1,2,3,4\}$. Suppose that
$$S_1=A'\cap B'\cap C',\quad S_2=A\cap B',\quad S_3=B'\cap C,\quad S_4=A'\cap B\cap C'.$$
Find the number of possible ordered tuples of sets of the form $(A\cap B,B\cap C,S_1,S_2,S_3,S_4)$.

In the diagram below, a smaller circle has its centre $O$ on the circumference of a bigger circle. The smaller circle intersects the bigger circle at $P$ and $Q$. Chord $PR$ of the smaller circle is tangent to the bigger circle. Line $RO$ is extended to intersect the bigger circle at $S$. Given that $\angle PRS=20^\circ$ and $\angle SPQ=x^\circ$, find $x$.

Given that $A=\sum_{i=0}^{2025}\binom{2025}{i}^2$, $B=\sum_{i=0}^{2024}\binom{2024}{i}^2$, find $\lfloor A/B\rfloor$.