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Explore the Back-StoryFind, with proof, all possible values of $t$ such that

$$\lim _{n \rightarrow \infty}{\frac{1+2^{1 / 3}+3^{1 / 3}+\cdots+n^{1 / 3}}{n^t}}=c$$

for some real number $c>0$. Also find the corresponding values of $c$.

Suppose $n \geq 2$. Consider the polynomial

$$

Q_n(x)=1-x^n-(1-x)^n .

$$

Show that the equation $Q_n(x)=0$ has only two real roots, namely 0 and 1.

Let $A B C D$ be a quadrilateral with all internal angles $<\pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta_1, \Delta_2, \Delta_3$ and $\Delta_4$ denote the areas of the shaded triangles shown. Prove that

$$

\Delta_1-\Delta_2+\Delta_3-\Delta_4=0

$$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which is differentiable at 0 . Define another function $g: \mathbb{R} \rightarrow \mathbb{R}$ as follows:

$$

g(x)= \begin{cases}f(x) \sin \left(\frac{1}{x}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases}

$$

Suppose that $g$ is also differentiable at 0 . Prove that

$$

g^{\prime}(0)=f^{\prime}(0)=f(0)=g(0)=0 .

$$

Let $P(x)$ be a polynomial with real coefficients. Let $\alpha_1, \ldots, \alpha_k$ be the distinct real roots of $P(x)=0$. If $P^{\prime}$ is the derivative of $P$ show that for each $i=1,2, \ldots, k$,

$$

\lim _{x \rightarrow a_i} \frac{\left(x-\alpha_i\right) P^{\prime}(x)}{P(x)}=r_i,

$$

for some positive integer $r_i$.

Q6. Let $x_1, \ldots, x_{2024}$ be non-negative real numbers with $\sum_{i=1}^{2024} x_i=1$. Find, with proof, the minimum and maximum possible values of the expression

$$

\sum_{i=1}^{1012} x_i+\sum_{i=1013}^{2024} x_i^2 .

$$

Consider a container of the shape obtained by revolving a segment of the parabola $x=1+y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1 \mathrm{~cm}^3 / \mathrm{s}$ into the container. Let $h(t)$ be the height of water inside the container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.

In a sports tournament involving $N$ teams, each team plays every other team exactly once. At the end of every match, the winning team gets 1 point and the losing team gets 0 points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:

$$

x_1 \geq x_2 \geq \cdots \geq x_N .

$$

Prove that for any $1 \leq k \leq N$,

$$

\frac{N-k}{2} \leq x_k \leq N-\frac{k+1}{2}

$$

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Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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