Found problems: 7
2024 ISI Entrance UGB, P3
Let $ABCD$ be a quadrilateral with all the internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4$ denote the areas of the shaded triangles as shown. Prove that \[\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.\]
[asy]
//made from sweat and hardwork by SatisfiedMagma
import olympiad;
import geometry;
size(250);
pair A = (-3,0);
pair B = (0,2);
pair C = (5.88,0.44);
pair D = (0.96, -1.86);
pair H = B + rotate(90)*(C-B);
pair G = C + rotate(270)*(B-C);
pair J = C + rotate(90)*(D-C);
pair I = D + rotate(270)*(C-D);
pair L = D + rotate(90)*(A-D);
pair K = A + rotate(270)*(D-A);
pair F = A + rotate(90)*(B-A);
pair E = B + rotate(270)*(A-B);
draw(B--H--G--C--B, blue);
draw(C--J--I--D--C, red);
draw(B--E--F--A--B, orange);
draw(D--L--K--A--D, magenta);
draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia);
pen lightFuchsia = deepgreen + 0.5*white;
fill(D--L--I--cycle, lightFuchsia);
fill(A--K--F--cycle, lightFuchsia);
fill(E--B--H--cycle, lightFuchsia);
fill(C--J--G--cycle, lightFuchsia);
label("$\triangle_2$", (E+B+H)/3);
label("$\triangle_4$", (D+L+I)/3);
label("$\triangle_3$", (C+G+J)/3);
label("$\triangle_1$", (A+F+K)/3);
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, S);
dot("$D$", D, N);
dot("$H$", H, dir(H));
dot("$G$", G, dir(G));
dot("$J$", J, dir(J));
dot("$I$", I, dir(I));
dot("$L$", L, dir(L));
dot("$K$", K, dir(K));
dot("$F$", F, dir(F));
dot("$E$", E, dir(E));
[/asy]
2024 ISI Entrance UGB, P1
Find, with proof, all possible values of $t$ such that
\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]
for some real $c>0$. Also find the corresponding values of $c$.
2024 ISI Entrance UGB, P4
Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows:
$$g(x) = \begin{cases}
f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\
0 &\text{if} ~ x = 0.
\end{cases}$$
Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]
2024 ISI Entrance UGB, P6
Let $x_1 , \dots , x_{2024}$ be non negative real numbers with $\displaystyle{\sum_{i=1}^{2024}}x_i = 1$. Find, with proof, the minimum and maximum possible values of the following expression \[\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .\]
2024 ISI Entrance UGB, P2
Suppose $n\ge 2$. 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 $1$.
2024 ISI Entrance UGB, P8
In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and 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: \[x_1 \ge x_2 \ge \cdots
\ge x_N . \]
Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]
2024 ISI Entrance UGB, P5
Let $P(x)$ be a polynomial with real coefficients. Let $\alpha_1 , \dots , \alpha_k$ be the distinct real roots of $P(x)=0$. If $P'$ is the derivative of $P$, show that for each $i=1,\dots , k$
\[\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i, \]
for some positive integer $r_i$.