Found problems: 85335
2003 National Olympiad First Round, 1
Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $A$ and $E$, what is $|AE|$?
$
\textbf{(A)}\ \dfrac 23
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \dfrac 32
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 3
$
1991 Kurschak Competition, 1
Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that
\[\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.\]
MBMT Guts Rounds, 2015.26
Choose a real number between $0$ and $10$, inclusive. If your number is less than the average of all numbers chosen, you will get your number's worth of points, but if your number is greater than or equal to the average, you will get $0$ points. For example, if the average of all numbers chosen is $1.2$, and you pick $1.6$, then you will receive $0$ points, but if you pick $0.5$, then you will receive $0.5$ points. Express your answer to the nearest thousandth. For example, $7.800$, $2.110$, and $0.234$ are valid responses, but $7.8$ and $0.2345$ are not. An invalid response will receive a score of zero.
2021 Peru MO (ONEM), 1
[b]a)[/b] Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.
[b]b)[/b] Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.
2017 IFYM, Sozopol, 2
With $\sigma (n)$ we denote the sum of the positive divisors of the natural number $n$. Prove that there exist infinitely many natural numbers $n$, for which $n$ divides $2^{\sigma (n)} -1$.
2008 Tournament Of Towns, 2
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?
2000 Austrian-Polish Competition, 3
For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$
2007 Purple Comet Problems, 1
The sum of nine consecutive odd numbers is $2007$. Find the greatest of these nine numbers.
2017 Princeton University Math Competition, 13
A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length $1$. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.
2024 Harvard-MIT Mathematics Tournament, 22
Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$
1993 APMO, 3
Let
\begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*}
be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.
2023 Harvard-MIT Mathematics Tournament, 10
The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.
2020 Tournament Of Towns, 3
There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations?
Maxim Didin
2009 Stanford Mathematics Tournament, 1
The sum of all of the interior angles of seven polygons is $180\times17$. Find the total number of sides of the polygons.
2005 District Olympiad, 1
Let $A_1$, $A_2$, $\ldots$, $A_n$, $n\geq 2$ be $n$ finite sets with the properties
i) $|A_i| \geq 2$, for all $1\leq i \leq n$;
ii) $|A_i\cap A_j| \neq 1$, for all $1\leq i<j\leq n$.
Prove that the elements of the set $\displaystyle \bigcup_{i=1}^n A_i$ can be colored with 2 colors, such that all the sets $A_i$ are bi-color, for all $1\leq i \leq n$.
MOAA Accuracy Rounds, 2023.8
Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow:
If an edge is numbered even, then the sum of the numbers of its endpoints must also be even.
If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd.
Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$.
[i]Proposed by Harry Kim[/i]
2013 NZMOC Camp Selection Problems, 10
Find the largest possible real number $C$ such that for all pairs $(x, y)$ of real numbers with $x \ne y$ and $xy = 2$, $$\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.$$ Also determine for which pairs $(x, y)$ equality holds.
1994 Polish MO Finals, 1
Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.
2010 May Olympiad, 4
Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.
2017 Romanian Masters In Mathematics, 1
[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers.
This number $k$ is called [i]weight[/i] of $n$.
[b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.
2010 239 Open Mathematical Olympiad, 6
We call natural numbers $n$ and $k$ are similar if they are multiples of square of a number greater than $1$. Let $f(n)$ denote the number of numbers from $1$ to $n$ similar to $n$ (for example, $f(16)=4$, since the number $16$ is similar to $4$, $8$, $12$ and $16$). What integer values can the quotient $\frac{n}{f(n)}$ take?
2004 AMC 10, 7
On a trip from the United States to Canada, Isabella took $ d$ U.S. dollars. At the border she exchanged them all, receiving $ 10$ Canadian dollars for every $ 7$ U.S. dollars. After spending $ 60$ Canadian dollars, she had $ d$ Canadian dollars left. What is the sum of the digits of $ d$?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 7\qquad
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 9$
2022 CMIMC, 2.7 1.3
For a family gathering, $8$ people order one dish each. The family sits around a circular table. Find the number of ways to place the dishes so that each person’s dish is either to the left, right, or directly in front of them.
[i]Proposed by Nicole Sim[/i]
2018 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$.
*The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$.
[i]Proposed by Zack Chroman[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.1
It is known that the graph of the function $y =\frac{a-6x}{2+x}$ is centrally summetric to the graph of the function $y = \frac{1}{x}$ with respect to some point. Find the value of the parameter $a$ and the coordinates of the center of symmetry.