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Putnam Mathematical Competition

The Putnam Mathematical Competition is an annual mathematics competition for undergraduate students in the United States and Canada. It is one of the most prestigious mathematics competitions in the world, and is known for its challenging problems.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(0) = 0$ and $f(1) = 1$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = c$.

  1. True
  2. False
Question 2 Multiple Choice (Single Answer)

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is zero, then $A$ is not invertible.

  1. True
  2. False
Question 3 Multiple Choice (Single Answer)

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b = c$.

  1. True
  2. False
Question 4 Multiple Choice (Single Answer)

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 1$.

  1. True
  2. False
Question 5 Multiple Choice (Single Answer)

Let $A$ be a $3 imes 3$ matrix with real entries. If the eigenvalues of $A$ are all real, then $A$ is diagonalizable.

  1. True
  2. False
Question 6 Multiple Choice (Single Answer)

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a = b = c$.

  1. True
  2. False
Question 7 Multiple Choice (Single Answer)

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) > 1$.

  1. True
  2. False
Question 8 Multiple Choice (Single Answer)

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is nonzero, then $A$ is invertible.

  1. True
  2. False
Question 9 Multiple Choice (Single Answer)

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b + c = 0$.

  1. True
  2. False
Question 10 Multiple Choice (Single Answer)

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 2$.

  1. True
  2. False
Question 11 Multiple Choice (Single Answer)

Let $A$ be a $3 imes 3$ matrix with real entries. If the eigenvalues of $A$ are all distinct, then $A$ is diagonalizable.

  1. True
  2. False
Question 12 Multiple Choice (Single Answer)

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a^2 + b^2 = c^2$.

  1. True
  2. False
Question 13 Multiple Choice (Single Answer)

Let $f(x)$ be a continuous function on the interval $[0, 1]$. If $f(x) > 0$ for all $x$ in $[0, 1]$, then there exists a point $c$ in the interval $(0, 1)$ such that $f(c) = 3$.

  1. True
  2. False
Question 14 Multiple Choice (Single Answer)

Let $A$ be a $3 imes 3$ matrix with real entries. If the determinant of $A$ is equal to the product of its eigenvalues, then $A$ is diagonalizable.

  1. True
  2. False
Question 15 Multiple Choice (Single Answer)

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b + c = p$.

  1. True
  2. False
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