Found problems: 1415
2011 Romanian Master of Mathematics, 6
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$.
(Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.)
[i](Romania) Dan Schwarz[/i]
2004 AIME Problems, 12
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.
2012 China Team Selection Test, 3
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$.
For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.
2011 Albania National Olympiad, 1
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
MathLinks Contest 7th, 4.2
Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.
Today's calculation of integrals, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
2010 IberoAmerican, 1
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
2023 Assara - South Russian Girl's MO, 7
A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle $ABC$ lie on this parabola so that the bisector of angle $\angle BAC$ is parallel to the axis $Ox$ . Prove that the midpoint of the median drawn from vertex $A$ lies on the axis of the parabola.
2003 AMC 12-AHSME, 9
A set $ S$ of points in the $ xy$-plane is symmetric about the origin, both coordinate axes, and the line $ y \equal{} x$. If $ (2, 3)$ is in $ S$, what is the smallest number of points in $ S$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 16$
2004 AMC 12/AHSME, 21
The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the first quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$
2010 Today's Calculation Of Integral, 551
In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.
2004 China Second Round Olympiad, 2
In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$. The x-intercept of line $A_nB_n$ is $a_n$, and the x-coordinate of point $B_n$ is $b_n$, $n\in\mathbb{N}$. Prove that
(1) $a_n>a_{n+1}>4$, $n\in\mathbb{N}$;
(2) There is $n_0\in\mathbb{N}$, such that for any $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.
2000 Baltic Way, 16
Prove that for all positive real numbers $a,b,c$ we have
\[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]
2022 AMC 12/AHSME, 5
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?
$\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)$
2003 AMC 12-AHSME, 9
Let $ f$ be a linear function for which $ f(6)\minus{}f(2)\equal{}12$. What is $ f(12)\minus{}f(2)$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 36$
1984 AMC 12/AHSME, 18
A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is
A. $\sqrt{2} - 1$
B. $\frac{1}{2}$
C. $2 - \sqrt{2}$
D. 1
E. Not uniquely determined
1994 Vietnam National Olympiad, 2
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
2011 AMC 12/AHSME, 22
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320 \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$
2008 All-Russian Olympiad, 8
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
2010 BMO TST, 2
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
2014 PUMaC Geometry A, 7
Let $O$ be the center of a circle of radius $26$, and let $A$, $B$ be two distinct points on the circle, with $M$ being the midpoint of $AB$. Consider point $C$ for which $CO=34$ and $\angle COM=15^\circ$. Let $N$ be the midpoint of $CO$. Suppose that $\angle ACB=90^\circ$. Find $MN$.
2013 Today's Calculation Of Integral, 873
Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$
(1) Find the condition for which $C_1$ is inscribed in $C_2$.
(2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$.
(3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$.
60 point
2013 Argentina Cono Sur TST, 3
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
2000 AIME Problems, 2
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$
2003 Vietnam Team Selection Test, 2
Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)