Found problems: 85335
2000 Balkan MO, 3
How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges?
2003 Romania National Olympiad, 4
Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral.
[i]Dinu Şerbănescu[/i]
2021 AMC 10 Spring, 10
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
2013 Irish Math Olympiad, 3
The altitudes of a triangle $\triangle ABC$ are used to form the sides of a second triangle $\triangle A_1B_1C_1$. The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle $\triangle A_2B_2C_2$. Prove that $\triangle A_2B_2C_2$ is similar to $\triangle ABC$.
1985 AMC 8, 12
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $ 6.2$ cm, $ 8.3$ cm, and $ 9.5$ cm. The area of the square is
\[ \textbf{(A)}\ 24 \text{ cm}^2 \qquad
\textbf{(B)}\ 36 \text{ cm}^2 \qquad
\textbf{(C)}\ 48 \text{ cm}^2 \qquad
\textbf{(D)}\ 64 \text{ cm}^2 \qquad
\textbf{(E)}\ 144 \text{ cm}^2
\]
2007 Romania Team Selection Test, 4
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
2012 Nordic, 3
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.
2017 Greece National Olympiad, 2
Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions.
In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.
2022 Kyiv City MO Round 1, Problem 5
Find the smallest integer $n$ for which it's possible to cut a square into $2n$ squares of two sizes: $n$ squares of one size, and $n$ squares of another size.
[i](Proposed by Bogdan Rublov)[/i]
2011 AMC 8, 6
In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?
$ \textbf{(A)} 20 \qquad\textbf{(B)} 25 \qquad\textbf{(C)} 45 \qquad\textbf{(D)} 306 \qquad\textbf{(E)} 351$
1973 Czech and Slovak Olympiad III A, 6
Consider a square of side of length 50. A polygonal chain $L$ is given in the square such that for every point $P$ of the square there is a point $Q$ of the chain with the property $PQ\le 1.$ Show that the length of $L$ is greater than 1248.
2021 Putnam, A2
For every positive real number $x$, let
\[
g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.
\]
Find $\lim_{x\to \infty}\frac{g(x)}{x}$.
[hide=Solution]
By the Binomial Theorem one obtains\\
$\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \right)^{\frac{1}{r}}$\\
$=\lim_{r \to 0}(1+r)^{\frac{1}{r}}=\boxed{e}$
[/hide]
1969 IMO Shortlist, 13
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
2009 Cuba MO, 8
Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.
1996 China National Olympiad, 1
$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$.
2007 AMC 10, 10
Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$?
$ \textbf{(A)}\ \text{two parallel lines}\qquad
\textbf{(B)}\ \text{a parabola}\qquad
\textbf{(C)}\ \text{a circle}\qquad
\textbf{(D)}\ \text{a line segment}\qquad
\textbf{(E)}\ \text{two points}$
2000 Harvard-MIT Mathematics Tournament, 9
How many hexagons are in the figure below with vertices on the given vertices?
(Note that a hexagon need not be convex, and edges may cross!)
[img]https://cdn.artofproblemsolving.com/attachments/1/9/437add8a9225760e7059b8dc2d481d562a7da2.png[/img]
1985 Poland - Second Round, 3
Let $ L $ be the set of all polylines $ ABCDA $, where $ A, B, C, D $ are different vertices of a fixed regular $1985$ -gon. We randomly select a polyline from the set $L$. Calculate the probability that it is the side of a convex quadrilateral.
2021 China Second Round A2, 2
Find the maximum value of $M$, if you choose $10$ different real numbers randomly in $[1,M]$, there must be $3$ numbers $a<b<c$, satisfy $ax^2+bx+c=0$ has no real root.
2014 Purple Comet Problems, 11
How many subsets of $\{1,2,3,4,\dots,12\}$ contain exactly one prime number?
2001 Grosman Memorial Mathematical Olympiad, 4
The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?
2010 Greece Team Selection Test, 1
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
2014 Taiwan TST Round 3, 3
Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.
2021 Purple Comet Problems, 18
Three red books, three white books, and three blue books are randomly stacked to form three piles of three books each. The probability that no book is the same color as the book immediately on top of it is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2005 Croatia National Olympiad, 2
The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.