SMO 2025 Open Round 1

In this paper, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and $\lceil x\rceil$ denote the smallest integer not less than $x$. For example, $\lfloor5\rfloor=5$, $\lfloor2.8\rfloor=2$, $\lfloor-2.3\rfloor=-3$; $\lceil5\rceil$, $\lceil2.8\rceil=3$, $\lceil-2.3\rceil=-2$.

On a cartesian plane, the equation $|x^2-x|=k$ has 4 distinct solutions for $a\lt k\lt b$. Find the largest possible value of $\lfloor4(b-a)\rfloor$.

On a cartesian plane, the area of the region bounded by the graphs $x^2+y^2=1$ and $|x|+|y|=1$ is denoted by $A$. Find $\lfloor10A\rfloor$.

The points $U(1,1)$ and $P(t,\frac1t)$ lie on the curve with equation $y=\frac1x$. The tangents at $U$ and $P$ meet at the point $(\frac23,\frac43)$. Find the value of $10t$.

The 10 non-negative integers $x_1,x_2,x_3,\ldots,x_{10}$ satisfy the equation $\displaystyle\sum_{k=1}^{10}(kx_k)^3=3024$. Find the value of $\displaystyle\sum_{k=1}^{10}x_k$.

A right-angled triangle has integer side lengths $p,q,q+1$, where $p\lt q$ and inradius 2025. Find $p$.

Among all the possible pairs of real numbers $(x,y)$ which satisfy the equation
$$(x^3-y)^{2024}+(x-y)^{2026}=0$$
find the maximum possible value of $|x|+|y|$.

Let $y=\displaystyle\sum_{k=0}^{20}\binom{20}{k}^2$. Find the number of consecutive zeros at the back of the number $(20!)^2y$ when it is written using its decimal representation.

In the acute triangle $ABC$, $BH$ is the altitude from $B$, $D$ and $E$ are all the midpoints of $AB$ and $AC$ respectively. Let $F$ be the reflection of $H$ across $DE$. Suppose $\angle C=40^\circ$ and $\angle FAH=35^\circ$. Find the size of $\angle A$ in degrees.

Let $\omega$ be a non-real complex number such that $\omega^{2025}=1$. Find the value of $10+\omega+\omega^2+\ldots+\omega^{2024}$.

The largest possible value of $x^2+y^2$, where $x$ and $y$ are real numbers, subject to the constraint $(x-2)^2+(y-2)^2=1$, is $A$. Find the value of $\lfloor A-1\rfloor$.

In a cartesian space with three mutually perpendicular $x$-, $y$- and $z$-axes in a counter-clockwise sense, a plane mirror has equation $x+y+3z=24$. A ray of light along the line with equation
$$\mathbf r=\begin{pmatrix}6\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\-4\end{pmatrix},\quad\lambda\in\mathbb R$$
hits the plane mirror and is reflected along the light with equation
$$\mathbf r=\begin{pmatrix}5\\1\\6\end{pmatrix}+v\begin{pmatrix}3\\a\\b\end{pmatrix},\quad v\in\mathbb R.$$
Find the value of $|a+b|$.

If $X=\left(\dfrac{\sqrt5+\sqrt3-\sqrt2}{\sqrt2}\right)^8+\left(\dfrac{\sqrt5-\sqrt3+\sqrt2}{\sqrt2}\right)^8$, find $\lceil X\rceil$, given that $\sqrt6\approx2.4495$.

It is given that $\displaystyle\int_1^3\frac{\lfloor x\rfloor}{x}\,dx=\ln B$. Find $\lfloor B\rfloor$.

Three towns $X,Y$ and $Z$ lie on a plane with coordinates $(0,0)$, $(200,0)$ and $(0,300)$ respectively. There are 100, 200 and 300 students in town $X,Y$ and $Z$ respectively. A school is to be built on a grid point $(x,y)$, where $x$ and $y$ are both integers, such that the overall distance travelled by all the students is minimized. Find the value of $x+y$.

Let $x$ be a real number such that $x^3+(2x^3+x-3)^3=3$. Find the value of $2x^3$.

Let $x$ and $y$ be distinct real numbers such that $x^2=2y+5$ and $y^2=2x+5$. Find the value of $2x^2y^2-x^3-y^3$.

Consider all possible real numbers $a,b$ and $c$ such that $a+b+c=0$ and $a^2+b^2+c^2=12$. Find the largest possible value of $a^4+b^4+c^4$.

Zhan Zhao writes down 100 numbers on the board and calculates their product. He then increases each number by 1 and observes that the product remains the same. He increases the numbers in the same way again and again, the product remains the same. He performs this procedure $k$ times, each time having the same product. Find the greatest possible value of $k$.

Find $a_9$ if for all real values of $x$, we have
$$(2x+1)^{20}=(x+1)^{10}(a_0+a_1x+a_2x^2+\ldots+a_{10}x^{10})+b_0+b_1x+b_2x^2+\ldots+b_9x^9.$$

Determine the largest possible integer $x$ such that $x^2+15x+31$ is a square number.