SMO 2024 Senior Round 1

Find the largest positive integer $A$ such that $2^x+\dfrac{2025}{2^x}-A>0$ for all real numbers $x$

If $x=\dfrac1{\log_{\frac{2024}{2023}}7}+\dfrac1{\log_{\frac{2023}{2022}}7}+\dfrac1{\log_{\frac{2022}{2021}}7}$, find $7^x$.

Simplify $\dfrac{2024}{\sqrt{4+\sqrt{12}}}+\dfrac{2024}{\sqrt{4-\sqrt{12}}}$.

Suppose $x^{1/3}+12=y^{1/3}$ for some real numbers $x$ and $y$. Find the minimum possible value of $y-x$.

Find the largest possible value of $\dfrac{\sqrt2\cos(2x)}{\sin(x)+\cos(x)}$.

(A) $\frac12$

(B) 1

(C) $\frac32$

(D) 2

(E) $\frac52$

If $\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}=4$, find the value of $15x$.

Find the smallest positive integer $K$ such that
$$x^2-200x+y^2=0\quad\text{and}\quad x+y\le K.$$

Given that $\dfrac{\cos(x)}{\sin(3x)}-\dfrac{\sin(x)}{\cos(3x)}-2\cdot\dfrac{\sin(4x)}{\cos(6x)}=2024$, find the value of $\dfrac{\cos(10x)}{\sin(12x)}$.

Find the smallest positive integer $k$ such that the coefficient of $x^k$ in the expansion of $\left(5x^3+\dfrac1{\sqrt{x}}\right)^{2024}$ is not zero.

Let
$$P=(2024^2+1)(2024^{2^2}+1)(2024^{2^3}+1)\cdots(2024^{2^{10}}+1)\times2025+\frac1{2023}.$$
Find the smallest positive integer $N$ such that $N>\log_{2024}P$.

Let $\triangle ABC$ be a triangle with area $1000$. Let $M$ and $N$ be points on $AB$ and $AC$ respectively such that
$$AM:MB=3:2\quad\text{and}\quad AN:NC=7:3.$$
Let $X$ and $Y$ be the midpoints of $BN$ and $CM$ respectively. Find the area of $\triangle AXY$.

Find the largest positive integer $n\le10000$ such that $1+2024n^2$ is a perfect square.

In a tetrahedron $SABC$, the faces $SBC$ and $ABC$ are perpendicular to each other. The angles $\angle ASB$, $\angle BSC$, $\angle ASC$ are all $60^\circ$, and $SB=SC=4$. Find the square of the volume of the tetrahedron.

Let $a,b,c$ be the three real roots of the cubic equation
$$2x^3-4x^2-21x-8=0.$$
Given that
$$S=\frac1{ab+c-1}+\frac1{bc+a-1}+\frac1{ca+b-1}$$
is a rational number that can be expressed as a fraction in the lowest form $\frac mn$, find the value of $m^2+n^2$.

Consider the equation
$$\frac{\sqrt3-1}{\sin x}+\frac{\sqrt3+1}{\cos x}=4\sqrt2.$$
For the range $0\lt x\lt \pi/2$, the sum of the solutions of the equation can be expressed in the form $\frac{m\pi}{n}$, where $\frac mn$ is a fraction in the lowest form. Find $m+n$.

An engineer constructs a circle with centre $O$ and diameter $CD$ on level ground, and builds a vertical tower of height $20$ at the centre. $B$ is another point on the circumference and $P$ is on $CD$ produced such that $PB$ is a secant line of the circle. Given that $PB=33$, $PC=77$ and $CD=74$, find the minimum possible distance of any point on $PB$ to the top of the tower.

$P$ is a common point of tangency of two circles. $BA$ is a chord of the larger circle which is tangent to the smaller circle at a point $C$. $PB$ and $PA$ intersect the smaller circle at points $E$ and $D$ respectively. If $BA=15$, $PE=2$, and $PD=3$, find the length $CA$.

On each face of a cube, an integer greater than $2$ is written. Each vertex of the cube is the intersection of three unique faces, and each edge is the intersection of two unique faces. Assign to each vertex the product of the numbers written on the faces intersecting the vertex, and assign to each edge the product of the numbers written on the faces intersecting the edge. The sum of the numbers assigned to the eight vertices is equal to $2024$. Find the maximum possible value of an edge.

Find the sum of the squares of each of the roots of the equation $$x^2-4\lfloor x\rfloor-12=0,$$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Calculate the remainder when $1901^{2024}$ is divided by $1216$.