Found problems: 85335
2002 AMC 10, 25
When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
1991 Austrian-Polish Competition, 7
For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge 0$ and find all positive $x$ for which the maximum is attained.
2003 Greece National Olympiad, 4
On the set $\Sigma$ of points of the plane $\Pi$ we define the operation $*$ which maps each pair $(X, Y )$ of points in $\Sigma$ to the point $Z = X * Y$ that is symmetric to $X$ with respect to $Y .$ Consider a square $ABCD$ in $\Pi$. Is it possible, using the points $A, B, C$ and applying the operation $*$ finitely many times, to construct the point $D?$
PEN R Problems, 2
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
Cono Sur Shortlist - geometry, 2012.G4.2
2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.
MathLinks Contest 2nd, 3.2
Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines $AD, BE, CF$ respectively, such that $$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$
Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.
2005 Silk Road, 1
Let $n \geq 2$ be natural number.
Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.
2022 JBMO Shortlist, C5
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$.
Proposed by [i]Viktor Simjanoski, Macedonia[/i]
2009 Tuymaada Olympiad, 2
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?
[i]Proposed by A. Golovanov[/i]
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 $1996n$.