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Explore the Back-StoryThis collection of problems and solutions from CMI Entrance 2022 is a work in progress. If you remember the problems, let us know in the comment section.

Let $a_0 , a_1, a_2…..$ be an arithmetic progression such that $a_0$ and $a_1$ are positive integers. Let $g_1, g_2, g_3,…$ be a geometric progression such that $g_0=a_0$ and $g_1=a_1$.

(i) $(a_5)^2 \geq a_0 a_{10}$

(ii) the sum $a_0+a_1+a_2+….+a_{10}$ is an integer multiple of $a_5$.

(iii) If $\sum_{i=0}^{\infty} a_i \to \infty $ then $\sum_{i=0}^{\infty} g_i \to \infty $

(iv) If $\sum_{i=0}^{\infty} g_i $ is finite , then $\sum_{i=0}^{\infty} a_i \to -\infty $

Let $A = \begin{bmatrix} 1 & 2 & 3\\ 10 & 20 & 30 \\ 11 & 22 & k \end{bmatrix}$ and $V = \begin{bmatrix} x\\ y \\ z \end{bmatrix}$.

(i) Matrix $A$ is not going be invertible regardless of the value of $k$.

(iii) $Av=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix} $ then the locus is a line or a plane containing the origin.

(iv) If $Av=\begin{bmatrix} p\\ q \\r \end{bmatrix}$ then $q=10p$

(I) $a= \frac{1}{\ln (3)}$ then $3^a=e$

(II) $\sin(0.02) < 2 \sin (0.01)$

(III) $\arctan (0.01) > 0.01 $

(IV) $\int_0^1 \arctan(x) dx = \pi - \ln 4$

Let $f(x)$ be a function whose domain is $[0,1]$.

I. $f $ is differentiable at each point in $[0,1]$

II. $f$ is continuous at each point in $[0,1]$.

III. The set $\{f(x) | x \in [0,1]\}$ has a minimum and a maximum element.

(i) I implies II

(ii) II implies III

(iii) Something something

(iv) No two statements are equivalent

We have the cards numbered from 1 to 9 arranged in a certain order. A *move* is defined as interchanging a numbered cardx with the card labelled 1. An arrangement is said to be *rectifiable* if it is possible to arrange the cards in descending order by a sequence of moves.

(1) Consider the premutation $9,1,2,3,4,5,6,7,8$. Atleast 8 or more moves required to get in the orginal order.

(IV) There exists a sequence of $1, 2, …, 9$ which is not rectifiable?

Let $z=a+ib$ ($a$, $b$ real number). Define $f(z)=z^{222}+ \frac{1}{z^{222}}$.

i) If $|z| =1$ then $f(z)$ real.

ii) If $z+ \frac{1}{z}=1$ then $f(z)=2$

If $\frac{WZ}{XY} = \frac{QZ}{YQ} = \frac{WP}{XP} = k$ show that $XP = XR$.

In $XY$ plane , grids are drawn.

i)A line $L$ is drawn from $(0,0)$ to $(m,n)$ then find the number of small squares $(1 \times 1)$ line $L$ will intersect. For example $(2,3)$ will intersect in $4$ squares.

ii) In a $n \times n$ square what is the maximum possible number of squares can be intersected by a line segment.

Consider $f(x)= 1 + x + x^2 + …. + x^n$

Find the number of local minima of $f$.

For each $c$ such that $(c,f(c))$ is a point of maximum or minimum, specify the integer $k$ such that $k \leq c < (k + 1) $

$(r,s)$ is said to be a good pair if $r$ and $s$ are distinct and $r^3+s^2=s^3+r^2$.

(i) Find good pair $(a,l)$ for the largest possible value of $l$. Also , find good pair $(s,b)$ for the smallest possible value of $s$. Also, prove that for every good pair $(c,d)$ other than the two mentioned above , there exists $e$ such that $(c,e)$ and $(d,e)$ are good pairs.

(ii) Show that there are infinitely many good pairs $(r,s)$ such that $r$ and $s$ are rational.

(i) Prove that $f(n)=n^2+n-1$ can have at most 2 roots modulo p. Where p is prime

(ii) Find the number of roots of f(n) mod 121

(iii) What can you tell about the cardinality of the set of roots of $f(n) \mod p^2$

This collection of problems and solutions from CMI Entrance 2022 is a work in progress. If you remember the problems, let us know in the comment section.

Let $a_0 , a_1, a_2…..$ be an arithmetic progression such that $a_0$ and $a_1$ are positive integers. Let $g_1, g_2, g_3,…$ be a geometric progression such that $g_0=a_0$ and $g_1=a_1$.

(i) $(a_5)^2 \geq a_0 a_{10}$

(ii) the sum $a_0+a_1+a_2+….+a_{10}$ is an integer multiple of $a_5$.

(iii) If $\sum_{i=0}^{\infty} a_i \to \infty $ then $\sum_{i=0}^{\infty} g_i \to \infty $

(iv) If $\sum_{i=0}^{\infty} g_i $ is finite , then $\sum_{i=0}^{\infty} a_i \to -\infty $

Let $A = \begin{bmatrix} 1 & 2 & 3\\ 10 & 20 & 30 \\ 11 & 22 & k \end{bmatrix}$ and $V = \begin{bmatrix} x\\ y \\ z \end{bmatrix}$.

(i) Matrix $A$ is not going be invertible regardless of the value of $k$.

(iii) $Av=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix} $ then the locus is a line or a plane containing the origin.

(iv) If $Av=\begin{bmatrix} p\\ q \\r \end{bmatrix}$ then $q=10p$

(I) $a= \frac{1}{\ln (3)}$ then $3^a=e$

(II) $\sin(0.02) < 2 \sin (0.01)$

(III) $\arctan (0.01) > 0.01 $

(IV) $\int_0^1 \arctan(x) dx = \pi - \ln 4$

Let $f(x)$ be a function whose domain is $[0,1]$.

I. $f $ is differentiable at each point in $[0,1]$

II. $f$ is continuous at each point in $[0,1]$.

III. The set $\{f(x) | x \in [0,1]\}$ has a minimum and a maximum element.

(i) I implies II

(ii) II implies III

(iii) Something something

(iv) No two statements are equivalent

We have the cards numbered from 1 to 9 arranged in a certain order. A *move* is defined as interchanging a numbered cardx with the card labelled 1. An arrangement is said to be *rectifiable* if it is possible to arrange the cards in descending order by a sequence of moves.

(1) Consider the premutation $9,1,2,3,4,5,6,7,8$. Atleast 8 or more moves required to get in the orginal order.

(IV) There exists a sequence of $1, 2, …, 9$ which is not rectifiable?

Let $z=a+ib$ ($a$, $b$ real number). Define $f(z)=z^{222}+ \frac{1}{z^{222}}$.

i) If $|z| =1$ then $f(z)$ real.

ii) If $z+ \frac{1}{z}=1$ then $f(z)=2$

If $\frac{WZ}{XY} = \frac{QZ}{YQ} = \frac{WP}{XP} = k$ show that $XP = XR$.

In $XY$ plane , grids are drawn.

i)A line $L$ is drawn from $(0,0)$ to $(m,n)$ then find the number of small squares $(1 \times 1)$ line $L$ will intersect. For example $(2,3)$ will intersect in $4$ squares.

ii) In a $n \times n$ square what is the maximum possible number of squares can be intersected by a line segment.

Consider $f(x)= 1 + x + x^2 + …. + x^n$

Find the number of local minima of $f$.

For each $c$ such that $(c,f(c))$ is a point of maximum or minimum, specify the integer $k$ such that $k \leq c < (k + 1) $

$(r,s)$ is said to be a good pair if $r$ and $s$ are distinct and $r^3+s^2=s^3+r^2$.

(i) Find good pair $(a,l)$ for the largest possible value of $l$. Also , find good pair $(s,b)$ for the smallest possible value of $s$. Also, prove that for every good pair $(c,d)$ other than the two mentioned above , there exists $e$ such that $(c,e)$ and $(d,e)$ are good pairs.

(ii) Show that there are infinitely many good pairs $(r,s)$ such that $r$ and $s$ are rational.

(i) Prove that $f(n)=n^2+n-1$ can have at most 2 roots modulo p. Where p is prime

(ii) Find the number of roots of f(n) mod 121

(iii) What can you tell about the cardinality of the set of roots of $f(n) \mod p^2$

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A11: (iii) if |z|=1 then |f(z)|≤2

In B3 we had to find both minima and maxima

In A11.. (iii) z + 1/z is Real then | f(z) | ≤ 2