Found problems: 85335
2023 Harvard-MIT Mathematics Tournament, 6
For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=r(111111001112)=111001111112=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_1, \ldots, a_8$ of odd positive integers such that $r(a_1), \ldots , r(a_8)$ is an arithmetic progression in that order.
1973 Spain Mathematical Olympiad, 7
The two points $P(8, 2)$ and $Q(5, 11)$ are considered in the plane. A mobile moves from $P$ to $Q$ according to a path that has to fulfill the following conditions: The moving part of $ P$ and arrives at a point on the $x$-axis, along which it travels a segment of length $1$, then it departs from this axis and goes towards a point on the $y$ axis, on which travels a segment of length $2$, separates from the $y$ axis finally and goes towards the point $Q$. Among all the possible paths, determine the one with the minimum length, thus like this same length.
2008 Argentina Iberoamerican TST, 3
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$
2020 USMCA, 18
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide what time. Each of the four group members sign on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
2013 India IMO Training Camp, 1
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.
2018 Peru Iberoamerican Team Selection Test, P5
Find all positive integers $a, b$, and $c$ such that the numbers
$$\frac{a+1}{b}, \frac{b+1}{c} \quad \text{and} \quad \frac{c+1}{a}$$
are positive integers.
2004 Thailand Mathematical Olympiad, 14
Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.
2020-IMOC, A3
$\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.}$ Assume that $a, b, c$ are positive reals such that $a + b + c = 3$. Prove that $$\definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.$$
[i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b].
[color=#3D9186]#1734[/color]
2008 Indonesia TST, 1
A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$
2000 Tournament Of Towns, 2
Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles.
(A Shapovalov)
2022 Switzerland Team Selection Test, 7
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
2002 Moldova National Olympiad, 4
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that:
$ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$
2022-2023 OMMC FINAL ROUND, 6
Evan writes a random positive integer on a board: the integer $k$ has probability $2^{-k}$ of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote $5,8,2,3,6,10,2$ in that order then his two sequences would be $5,8,2,3,6,10$ and $2,3,5,6,8,10.$
Find the probability that for all $k \in \{ 1,4,34 \},$ that $k$ was written, and $k$ appears in the same position in both sequences.
2019 CHMMC (Fall), 7
Let $S$ be the set of all positive integers $n$ satisfying the following two conditions:
$\bullet$ $n$ is relatively prime to all positive integers less than or equal to $\frac{n}{6}$
$\bullet$ $2^n \equiv 4$ mod $n$
What is the sum of all numbers in $S$?
2015 Auckland Mathematical Olympiad, 3
In the calculation $HE \times EH = WHEW$, where different letters stand for different nonzero digits. Find the values of all the letters.
1996 German National Olympiad, 1
Find all natural numbers $n$ with the following property:
Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.
2015 Cuba MO, 4
Let $A$ and $B$ be two subsets of $\{1, 2, 3, 4, ..., 100\}$, such that $|A| = |B|$ and $A\cap B =\emptyset$. If $n \in A$ implies that $2n + 2 \in B$, determine the largest possible value of $ |A \cup B|$.
1990 National High School Mathematics League, 5
Two non-zero-complex numbers $x,y$, satisfy that $x^2+xy+y^2=0$. Then the value of $(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}$ is
$\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}$none above
2013 Stanford Mathematics Tournament, 23
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.
2018 Greece Junior Math Olympiad, 3
Let $a$ and $b$ be positive integers with $b$ odd, such that the number $$\frac{(a+b)^2+4a}{ab}$$ is an integer. Prove that $a$ is a perfect square.
2021 LMT Spring, B10
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$.
[i]Proposed by Zachary Perry[/i]
2014 Online Math Open Problems, 29
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
[i]Proposed by Evan Chen[/i]
2016 South African National Olympiad, 2
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations
$$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$
are four consecutive integers.
2021 South East Mathematical Olympiad, 6
Let $ABCD$ be a cyclic quadrilateral. The internal angle bisector of $\angle BAD$ and line $BC$ intersect at $E.$ $M$ is the midpoint of segment $AE.$ The exterior angle bisector of $\angle BCD$ and line $AD$ intersect at $F.$ The lines $MF$ and $AB$ intersect at $G.$ Prove that if $AB=2AD,$ then $MF=2MG.$
2017 Iran MO (2nd Round), 5
There are five smart kids sitting around a round table. Their teacher says: "I gave a few apples to some of you, and none of you have the same amount of apple. Also each of you will know the amount of apple that the person to your left and the person to your right has."
The teacher tells the total amount of apples, then asks the kids to guess the difference of the amount of apple that the two kids in front of them have.
$a)$ If the total amount of apples is less than $16$, prove that at least one of the kids will guess the difference correctly.
$b)$ Prove that the teacher can give the total of $16$ apples such that no one can guess the difference correctly.