Found problems: 85335
2008 Purple Comet Problems, 14
A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $
1989 AMC 12/AHSME, 18
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all:
$\textbf{(A)}\ \text{ integers } x \qquad
\textbf{(B)}\ \text{ rational } x \qquad
\textbf{(C)}\ \text{ real } x\qquad
\textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad
\textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$
2011 Saudi Arabia Pre-TST, 2
Find all positive integers $x$ and $y$ such that $${x \choose y} = 1432$$
TNO 2024 Junior, 5
The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.
2023 VIASM Summer Challenge, Problem 2
Given a $20 \times 101$ square table with $20$ rows and $101$ columns. One wants to fill numbers $0$ and $1$ in the unit squares of the table satisfying the following conditions:
$[\text{i}]$ Each square has exactly one number to be filled in.
$[\text{ii}]$ Each column is filled with exactly two $1'$s.
$[\text{iii}]$ Any two rows with no more than one column are filled with two $1'$s.
$a.$ How many ways to fill the numbers satisfying the given conditions?
$b.$ With a satisfied numbering way, we number the rows in order from top to bottom. A triple of row (distinct, unordered) $\{a; b; c\}$ is said to be [i]united[/i] if the sets of numbers in the three rows are $(a_1, a_2, . . . , a_{101}), (b_1, b_2, . . . . , b_{101}),$ and $(c_1, c_2, . . . . , c_{101})$ satisfied$$\sum\limits_{i = 1}^{101} {({a_i}{b_i} + {b_i}{c_i} + {c_i}{a_i})} = 3.$$
Prove that: there are at least $10$ [i]united[/i] sets.
2010 National Olympiad First Round, 17
Let $A,B,C,D$ be points in the space such that $|AB|=|AC|=3$, $|DB|=|DC|=5$, $|AD|=6$, and $|BC|=2$. Let $P$ be the nearest point of $BC$ to the point $D$, and $Q$ be the nearest point of the plane $ABC$ to the point $D$. What is $|PQ|$?
$ \textbf{(A)}\ \frac{1}{\sqrt 2}
\qquad\textbf{(B)}\ \frac{3\sqrt 7}{2}
\qquad\textbf{(C)}\ \frac{57}{2\sqrt{11}}
\qquad\textbf{(D)}\ \frac{9}{2\sqrt 2}
\qquad\textbf{(E)}\ 2\sqrt 2
$
2025 China Team Selection Test, 11
Let \( n \geq 4 \). Proof that
\[
(2^x - 1)(5^x - 1) = y^n
\]
have no positive integer solution \((x, y)\).
2005 China Team Selection Test, 2
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
2020 Abels Math Contest (Norwegian MO) Final, 3
Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .
2002 National High School Mathematics League, 6
Consider the area encircled by $x^2=4y,x^2=-4y,x=4,x=-4$, rotate it around $y$-axis, the volume of the revolved body is $V_1$. Then consider the figure formed by all points $(x,y)$ that $x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4$, rotate it around $y$-axis, the volume of the revolved body is $V_2$. The relationship between $V_1$ and $V_2$ is
$\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2$