Practice (131)

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Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.

If $2^n-1$ is a prime number, prove $n$ must be a prime number too.

There are $7$ boys each of which has at least $3$ brothers among the other $6$ boys. Are these $7$ boys necessarily all brothers? Explain.

If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.

Show that if $a$, $b$ and $c$ are odd integers, then the equation $ax^2 + bx + c=0$ has no integer solution.

Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.

A prime number is called an absolute prime if every permutation of its digits in base 10 is also a prime number. For example: 2, 3, 5, 7, 11, 13 (31), 17 (71), 37 (73) 79 (97), 113 (131, 311), 199 (919, 991) and 337 (373, 733) are absolute primes. Prove that no absolute prime contains all of the digits 1, 3, 7 and 9 in base 10.

Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?


Let $p_1, p_2, \cdots, p_n$ be distinct integers and let $f(x)$ be the polynomial of degree $n$ given by $$f(x) = (x - p_1)(x - p_2)\cdots (x -p_n)$$ Prove that the polynomial $g(x) = (f(x))^2 + 1$ cannot be expressed as the product of two non-constant polynomials with integral coefficients.


Is it possible to write the polynomial $f(x) = x^{105}-9$ as the product of two polynomials of degree less than 105 with integer coefficients?

Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.

Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

The sum of two positive integers is $2310$. Show that their product is not divisible by $2310$.

Is it possible for a geometric sequence to contain three distinct prime numbers?

Let the lengths of five line segments be $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$, respectively, where $a_1 \ge a_2\ge a_3\ge a_4\ge a_5$. If any three of these five line segments can form a triangle, then prove at least one of such triangle is acute.

Given $n > 2$ points on a plane. Prove if any straight line passing two of these points, it must pass another one among these points, then all these $n$ points must be collinear.


If all sides of a convex pentagon $ABCDE$ are equal in length and $\angle{A}\ge\angle{B}\ge\angle{C}\ge\angle{D}\ge\angle{E}$, show that $ABCDE$ is a regular pentagon.

(Bezout's theorem) Show that two positive integers $a$ and $b$ are co-prime if there exist integer $x$ and $y$ satisfying $ax+by=1$.

Show there exist infinite many primes in the form of $(4k+1)$ where $k$ is a positive integer.

Find all pairs of positive integers $(a, b)$ satisfying $a! + b! = a^b + b^a$.