Practice (97)
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Given $S = \{2, 5, 8, 11, 14, 17, 20,\cdots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1, A2,\cdots A_n\}$, s.t. $\displaystyle\sum_{i=1}^{n}\frac{1}{A_i}=1$ Find the minimum possible value of $n$.
Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?
How many ordered pairs of integers $(x,y)$ are there such that $x^2 + 2xy+3y^2=34$?
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?
A grid point is defined as a point whose $x$ and $y$ coordinates are both integers. How many grid points are there on the circle which is centered at (199, 0) with a radius of 199?
For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?
A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?
Determine all positive integers $m$ and $n$ such that $m^2+1$ is a prime number and $10(m^2 + 1) = n^2 + 1$.
Find all right triangles whose sides' lengths are all integers, and areas equal circumstance numerically.
Solve in positive integers the equation $3^x + 4^y = 5^z$ .
Solve in positive integers the equation $8^x + 15^y = 17^z$.
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 16n = 2016$
Let $a, b$, and $c$ be three positive integers such that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$. Find the sum of all possible $a$ where $a \le 100$.
There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.
The sum of the three different positive unit fractions is $\frac{6}{7}$. What is the least number that could be the sum of the denominators of these fractions?
Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.
In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{a_k^2+b_k^2}$ where $\{a_k\}$ is an arithmetic sequence whose first term is $4$ and common difference is $8$. $b_1, b_2,\cdots, b_n$ are positive real numbers satisfying $\displaystyle\sum_{k=1}^nb_k=17$. If there exist a positive integer $n$ such that $S_n$ is also an integer, find $n$.
Find all postive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.
Proposed by Serbia
In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining
medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.
Show that neither $385^{97}$ nor $366^{17}$ can be expressed as the sum of cubes of some consecutive integers.
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
Consider the equation \[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]
(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.
(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.
Let integers $u$ and $v$ be two integral roots to the quadratic equation $x^2 + bx+c=0$ where $b+c=298$. If $u < v$, find the smallest possible value of $v-u$.