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CMI 2023 | Mathematics | Part A | Question: 5
Consider the real matrix\[A=\left(\begin{array}{ll}\lambda & 2 \\3 & 5\end{array}\right) .\]Assume that $-1$ is an eigenvalue of $A$. Which of the following are true?The ...
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CMI2023-B: 1
Let $\Sigma=\{a, b\}$ be an alphabet. A palindrome is a word which reads the same when read from left-to-right, or from right-to-left. For example, the words $a b b a, a ...
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CMI2023-B: 2
A non-empty collection $\mathcal{S}$ of subsets of a set $U$ is a scattering of $U$ if it satisfies the following condition: for all $A, B \i...
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CMI2023-B: 3
Let $a_{1}, \ldots, a_{n}$ be integers. Show that for some $k, m$ such that $1 \leq k \leq m \leq n$, the sum $a_{k}+a_{k+1}+\cdots+a_{m}$ is divisible by $n$. (Hint: Con...
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CMI2023-B: 4
On the island of Knights, Knaves, and Normals, Knights always tell the truth, Knaves always lie, and Normals sometimes tell the truth and lie sometimes.One day Professor ...
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CMI2023-B: 5
You are starting a new bus service. You are hiring drivers and conductors. A driver and a conductor can run a bus only if they can speak a common language. There are $n$ ...
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CMI2023-B: 6
The input to the problem consists of (i) an array $A[1,2, \ldots, n]$ of $n$ positive integers and (ii) a positive integer $T$. We are given the guarantee that at least o...
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CMI2023-B: 7
Consider the following code which computes a function f . The input to f is an array $\mathrm{A}[1 . \mathrm{m}]$ which represents a number N in ternary. For example, $\m...
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CMI2023-A: 1
A candy factory uses 5 fruit flavours $\{A, B, C, D, E\}$. Each candy is made using one or more of these flavours. The taste of a candy depends on which flavours are incl...
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CMI2023-A: 2
How many elements are in the following set?\[\{(A, B) \mid A \subseteq B \subseteq\{1,2,3, \ldots, n\}\}\]$2^{n-1}$$3^{n}$$2^{n+1}$$2^{2 n}$ 
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CMI2023-A: 3
At a kindergarten, $2024$ kids sit in a circle. Suddenly, each kid randomly pokes either the kid to their left or the one to their right with equal probability. What is t...
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CMI2023-A: 4
A graph is $k$-regular if all the vertices have degree exactly $k$. What is the minimum number of vertices in a $k$-regular graph that has no $3$ -length cycles?$k$$k+1$$...
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CMI2023-A: 5
In a connected graph, any two paths of maximum length:have at least one vertex in common, but not necessarily an edge in commonhave at least one edge in commonhave at lea...
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CMI2023-A: 6
Consider the automaton over the alphabet $\{a, b, c\}$ shown in Figure $1$. The initial state is the leftmost state. States with a double circle are accepting states. Wha...
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CMI2023-A: 7
Let $L$ be a regular language, and let $n=10$. Which of the following statements is true?$L \cup\left\{a^{n} b^{n}\right\}$ is regular$L \cup\left\{a^{n} b^{n}\right\}$ i...
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CMI2023-A: 8
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow \mathbb{N}$ be functions over the set $\mathbb{N}$ of natural numbers. We will say:$f(n)=O(g(n))...
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CMI2023-A: 9
We assume that all the arguments are non-negative integers. The operation x div $2$ divides x by $2$ and returns an integer, discarding the fractional part.int f(int x, i...
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CMI2023-A: 10
We assume that all the arguments are non-negative integers. The operation x div $2$ divides x by $2$ and returns an integer, discarding the fractional part.int f(int x, i...
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