Let $x_1, x_2, \ldots, x_n$ be non-negative real numbers such that $\sum_{i=1}^{n} x_i = 1$. What is the maximum possible value of $\sum_{i=1}^{n} \sqrt{x_i}$?
(A) 1
(B) $\sqrt{n}$
(C) $n^{3/4}$
(D) $n$

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
$$\sum _ { i = 1 } ^ { 1012 } x _ { i } + \sum _ { i = 1013 } ^ { 2024 } x _ { i } ^ { 2 }$$

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 $1 \leq k \leq N$,
$$\frac { N - k } { 2 } \leq x _ { k } \leq N - \frac { k + 1 } { 2 } .$$

Let $x$ be an irrational number. If $a , b , c$ and $d$ are rational numbers such that $\frac { a x + b } { c x + d }$ is a rational number, which of the following must be true?
(A) $a d = b c$
(B) $a c = b d$.
(C) $a b = c d$.
(D) $a = d = 0$

Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is differentiable and $\left| f ^ { \prime } ( x ) \right| < \frac { 1 } { 2 }$ for all $x \in \mathbb { R }$. Show that for some $x _ { 0 } \in \mathbb { R } , f \left( x _ { 0 } \right) = x _ { 0 }$.

Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.

Let $a , b , c$ be nonzero real numbers such that $a + b + c \neq 0$. Assume that $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = \frac { 1 } { a + b + c }$$ Show that for any odd integer $k$, $$\frac { 1 } { a ^ { k } } + \frac { 1 } { b ^ { k } } + \frac { 1 } { c ^ { k } } = \frac { 1 } { a ^ { k } + b ^ { k } + c ^ { k } }$$

Let $\mathbb { N }$ denote the set of natural numbers, and let $\left( a _ { i } , b _ { i } \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb { N } \times \mathbb { N }$. Show that there are three distinct elements in the set $\left\{ 2 ^ { a _ { i } } 3 ^ { b _ { i } } : 1 \leq i \leq 9 \right\}$ whose product is a perfect cube.

Consider a ball that moves inside an acute-angled triangle along a straight line, until it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence $=$ angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

Let $n \geq 2$ and let $a _ { 1 } \leq a _ { 2 } \leq \cdots \leq a _ { n }$ be positive integers such that $\sum _ { i = 1 } ^ { n } a _ { i } = \Pi _ { i = 1 } ^ { n } a _ { i }$. Prove that $\sum _ { i = 1 } ^ { n } a _ { i } \leq 2n$ and determine when equality holds.

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(a) $75 ^ { \circ }$.
(B) $85 ^ { \circ }$.
(C) $95 ^ { \circ }$.
(D) $110 ^ { \circ }$.

9. Prove that the equation:
$$\arctan ( x ) + x ^ { 3 } + e ^ { x } = 0$$
has one and only one real solution.

Let $ABC$ be a right triangle with right angle at $A$. Let $O$ be the center of the square $BCDE$ constructed on the hypotenuse, on the opposite side from vertex $A$.
Prove that $O$ is equidistant from the lines $AB$ and $AC$.

1. A triangle $A B C$ is given, right-angled at $B$. Prove that this triangle is isosceles if and only if the altitude $B H$ relative to the hypotenuse is congruent to half the hypotenuse.

4. Determine the domain of the function $f ( x ) = \ln \left( \frac { a x - 7 } { x ^ { 2 } } \right)$, with $a$ a positive real parameter. Subsequently, identify the value of $a$ for which the hypotheses of Rolle's theorem are satisfied on the interval [1; 7] and the coordinates of the point that verifies the conclusion.

6. In a Cartesian coordinate system $O x y$, consider the equilateral hyperbola with equation $x y = k$, with $k$ a non-zero real parameter. Let $t$ be the tangent line to the hyperbola at a point $P$ of it. Let $A$ and $B$ be the points where $t$ intersects the axes of the reference frame. Prove that the triangles $A P O$ and $B P O$ are equivalent and that their area does not depend on the choice of $P$.

1. Given a triangle $A B C$, let $M$ be the midpoint of side $B C$ and let $B ^ { \prime }$ and $C ^ { \prime }$ be two points, respectively, on side $A B$ and on side $A C$, such that $A B ^ { \prime } = \frac { 1 } { 3 } A B$ and $A C ^ { \prime } = \frac { 1 } { 3 } A C$. Prove that, if the segments $M B ^ { \prime }$ and $M C ^ { \prime }$ are congruent to each other, then so are the sides $A B$ and $A C$.

6. Let Cl and C 2 be, respectively, the parabolas $\mathrm { x } 2 = \mathrm { y } - 1$ and $\mathrm { y } 2 = \mathrm { a }$. Let P be any point on C 1 and Q be any point on C 2 . Let P 1 and Q 1 be the reflections of P and Q , respectively, with respect to the line y $= \mathrm { x }$. Prove that P 1 lies on $\mathrm { C } 2 , \mathrm { Q } 1$ lies on Cland PQ $\geq \min \{ \mathrm { PP } 1 , \mathrm { QQ } 1 \}$. Hence or otherwise, determine points P0 and Q on the parabolas Cl and C 2 respectively such that $\mathrm { P } 0 \mathrm { Q } \leq \mathrm { P } 0$ for all pairs of points $( \mathrm { P } , \mathrm { Q } )$ with P on C 1 and Q on C . 7.
(a)
$$\begin{aligned} & \text { Suppose } P ( x ) = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \ldots \ldots . . + a _ { n } x ^ { 7 } . \text { If } | p ( x ) | \leq \left| e ^ { x - 1 } - 1 \right| \text { for all } \\ & x \geq 0 \text {, prove that } \left| a _ { 1 } + 2 a _ { 2 } + \ldots \ldots . + n a _ { n } \right| \leq 1 \end{aligned}$$
(b) For $x > 0$, let $f ( x ) = \int _ { 1 } ^ { x } \frac { \ln t } { 1 + \tau } d t$. Find the function $f ( x ) + f ( 1 / x )$ and show that $f ( e ) + f ( 1 / e ) = 1 / 2$. Here $\ln t = \log _ { e } t$.

If f is an even function then prove that $$\int _ { 0 } ^ { \pi / 2 } f ( \cos 2 x ) \cos x d x = \sqrt { 2 } \int _ { 0 } ^ { \pi / 4 } f ( \sin 2 x ) \cos x d x$$

Using the relation $2 ( 1 - \cos x ) < x ^ { 2 } , x ^ { 1 } 0$ or otherwise, prove that $\sin ( \tan x ) > x \forall x \hat { \mathrm { I } } [ 0 , \pi / 4 ]$.

If $\mathrm { P } ( 1 ) = 0$ and $( \mathrm { dP } ( \mathrm { x } ) ) / \mathrm { dx } > \mathrm { P } ( \mathrm { x } )$ for all $\mathrm { x } > 1$ then prove that $\mathrm { P } ( \mathrm { x } ) > 0$ for all $\mathrm { x } > 1$.

If In is the area of $n$ sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that $$I _ { n } = \frac { O _ { n } } { 2 } \left( 1 + \sqrt { 1 - \left( \frac { 2 l _ { n } } { n } \right) ^ { 2 } } \right)$$

8. If $\mathrm { p } ( \mathrm { x } ) = 51 \mathrm { x } ^ { 101 } - 2323 \mathrm { x } ^ { 100 } - 45 \mathrm { x } + 1035$, using Rolle's Theorem, prove that atleast one root lies between ( $45 ^ { 1 / 100 } , 46$ ). Sol. Let $\mathrm { g } ( \mathrm { x } ) = \int \mathrm { p } ( \mathrm { x } ) \mathrm { dx } = \frac { 51 \mathrm { x } ^ { 102 } } { 102 } - \frac { 2323 \mathrm { x } ^ { 101 } } { 101 } - \frac { 45 \mathrm { x } ^ { 2 } } { 2 } + 1035 \mathrm { x } + \mathrm { c }$ $= \frac { 1 } { 2 } \mathrm { x } ^ { 102 } - 23 \mathrm { x } ^ { 101 } - \frac { 45 } { 2 } \mathrm { x } ^ { 2 } + 1035 \mathrm { x } + \mathrm { c }$. Now $\mathrm { g } \left( 45 ^ { 1 / 100 } \right) = \frac { 1 } { 2 } ( 45 ) ^ { \frac { 102 } { 100 } } - 23 ( 45 ) ^ { \frac { 101 } { 100 } } - \frac { 45 } { 2 } ( 45 ) ^ { \frac { 2 } { 100 } } + 1035 ( 45 ) ^ { \frac { 1 } { 100 } } + \mathrm { c } = \mathrm { c }$ $\mathrm { g } ( 46 ) = \frac { ( 46 ) ^ { 102 } } { 2 } - 23 ( 46 ) ^ { 101 } - \frac { 45 } { 2 } ( 46 ) ^ { 2 } + 1035 ( 46 ) + \mathrm { c } = \mathrm { c }$. So $\mathrm { g } ^ { \prime } ( \mathrm { x } ) = \mathrm { p } ( \mathrm { x } )$ will have atleast one root in given interval.

8. If $\mathrm { p } ( \mathrm { x } ) = 51 \mathrm { x } ^ { 101 } - 2323 \mathrm { x } ^ { 100 } - 45 \mathrm { x } + 1035$, using Rolle's Theorem, prove that atleast one root lies between ( $45 ^ { 1 / 100 } , 46$ ). Sol. Let $\mathrm { g } ( \mathrm { x } ) = \int \mathrm { p } ( \mathrm { x } ) \mathrm { dx } = \frac { 51 \mathrm { x } ^ { 102 } } { 102 } - \frac { 2323 \mathrm { x } ^ { 101 } } { 101 } - \frac { 45 \mathrm { x } ^ { 2 } } { 2 } + 1035 \mathrm { x } + \mathrm { c }$ $= \frac { 1 } { 2 } \mathrm { x } ^ { 102 } - 23 \mathrm { x } ^ { 101 } - \frac { 45 } { 2 } \mathrm { x } ^ { 2 } + 1035 \mathrm { x } + \mathrm { c }$. Now $\mathrm { g } \left( 45 ^ { 1 / 100 } \right) = \frac { 1 } { 2 } ( 45 ) ^ { \frac { 102 } { 100 } } - 23 ( 45 ) ^ { \frac { 101 } { 100 } } - \frac { 45 } { 2 } ( 45 ) ^ { \frac { 2 } { 100 } } + 1035 ( 45 ) ^ { \frac { 1 } { 100 } } + \mathrm { c } = \mathrm { c }$ $\mathrm { g } ( 46 ) = \frac { ( 46 ) ^ { 102 } } { 2 } - 23 ( 46 ) ^ { 101 } - \frac { 45 } { 2 } ( 46 ) ^ { 2 } + 1035 ( 46 ) + \mathrm { c } = \mathrm { c }$. So $\mathrm { g } ^ { \prime } ( \mathrm { x } ) = \mathrm { p } ( \mathrm { x } )$ will have atleast one root in given interval.