Found problems: 85335
2018 Latvia Baltic Way TST, P2
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
2016 India Regional Mathematical Olympiad, 4
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?
2006 Baltic Way, 13
In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if
$\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$
then the points $A,D,F,E$ lie on a circle.
1996 China Team Selection Test, 1
Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.
1983 Tournament Of Towns, (045) 2
Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.
1997 AIME Problems, 2
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 CHMMC Winter (2021-22), 2
A prefrosh is participating in Caltech’s “Rotation.” They must rank Caltech’s $8$ houses, which are Avery, Page, Lloyd, Venerable, Ricketts, Blacker, Dabney, and Fleming, each a distinct integer rating from $1$ to $8$ inclusive. The conditions are that the rating $x$ they give to Fleming is at most the average rating $y$ given to Ricketts, Blacker, and Dabney, which is in turn at most the average rating $z$ given to Avery, Page, Lloyd, and Venerable. Moreover $x, y, z$ are all integers. How many such rankings can the prefrosh provide?
2018 CMIMC Number Theory, 9
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\]
2013 Today's Calculation Of Integral, 896
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
1987 National High School Mathematics League, 5
Two sets $M=\{x,xy,\lg(xy)\},N=\{0,|x|,y\}$, if $M=N$, then $(x+\frac{1}{y})+(x^2+\frac{1}{y^2})+\cdots+(x^{2001}+\frac{1}{y^{2001}})=$________.
2010 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.
2010 Contests, 3
Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying
$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$
Is $N$ even or odd?
Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.
2009 Peru MO (ONEM), 4
Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?
2024 Kyiv City MO Round 2, Problem 3
$2024$ ones and $2024$ twos are arranged in a circle in some order. Is it always possible to divide the circle into
[b]a)[/b] two (contiguous) parts with equal sums?
[b]b)[/b] three (contiguous) parts with equal sums?
[i]Proposed by Fedir Yudin[/i]
2022 Germany Team Selection Test, 2
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection.
Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.
[i]Proposed by Warut Suksompong, Thailand[/i]
2018 Hanoi Open Mathematics Competitions, 3
There are $3$ unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/8963e6716b7d9b23479dd7e106b4bd9a3267c1.png[/img]
A. $48$ B. $96$ C. $108$ D. $192$ E. $216$
2024 UMD Math Competition Part I, #20
There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?\[
\mathrm a. ~4/15\qquad \mathrm b. ~2/7 \qquad \mathrm c. ~2/9 \qquad\mathrm d. ~1/3\qquad\mathrm e. ~1/5\qquad\]
1971 IMO Longlists, 18
Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that
\[(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.\]
2019 Spain Mathematical Olympiad, 4
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$
2013 BMT Spring, 4
Find the sum of all real numbers $x$ such that $x^2 = 5x + 6\sqrt{x} - 3$.
2017 HMNT, 4
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
1989 National High School Mathematics League, 9
Functions $f_0(x)=|x|,f_1(x)=|f_0(x)-1|,f_2(x)=|f_1(x)-2|$. Area of the closed part between the figure of $f_2(x)$ and $x$-axis is________.
2024 AMC 8 -, 13
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$
2002 AMC 12/AHSME, 14
For all positive integers $ n$, let $ f(n) \equal{} \log_{2002} n^2$. Let
\[ N \equal{} f(11) \plus{} f(13) \plus{} f(14)
\]
Which of the following relations is true?
$ \textbf{(A)}\ N < 1 \qquad \textbf{(B)}\ N \equal{} 1 \qquad \textbf{(C)}\ 1 < N < 2 \qquad \textbf{(D)}\ N \equal{} 2 \qquad \textbf{(E)}\ N > 2$
2018 Taiwan TST Round 1, 3
There are $n$ husbands and wives at a party in the palace. The husbands sit at a round table, and the wives sit at another round tables. The king and queen (not included in the $n$ couples) are going to shake hands with them one by one. Assume that the king starts from a man, and the queen starts from his wife. Consider the following two ways of shaking hands:
(i) The king shakes hands with the men one by one clockwise. Each time when the king shakes hands with a man, the queen moves clockwise to his wife and shakes hands with her. Assume that at last when the king gets back to the man he begins with, the queen goes around the table $a$ times.
(ii) The queen shakes hands with the women one by one clockwise. Each time when the queen shakes hands with a woman, the king moves clockwise to her husband and shakes hands with him. Assume that at last when the queen gets back to the woman she begins with, the king goes around the table $b$ times.
Determine the maximum possible value of $|a-b|$.