Found problems: 222
2002 Czech and Slovak Olympiad III A, 5
A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)
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}$
1969 AMC 12/AHSME, 11
Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals:
$\textbf{(A) }-\tfrac35\qquad
\textbf{(B) }-\tfrac25\qquad
\textbf{(C) }-\tfrac15\qquad
\textbf{(D) }\tfrac15\qquad
\textbf{(E) }\text{either }-\tfrac15\text{ or }\tfrac15$
2005 ISI B.Math Entrance Exam, 5
Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .
2020 AIME Problems, 2
Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1998 AMC 12/AHSME, 29
A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$ \textbf{(A)}\ 4.0\qquad
\textbf{(B)}\ 4.2\qquad
\textbf{(C)}\ 4.5\qquad
\textbf{(D)}\ 5.0\qquad
\textbf{(E)}\ 5.6$
1988 National High School Mathematics League, 3
On the coordinate plane, is there a line family of infinitely many lines $l_1,l_2,\cdots,l_n,\cdots$, satisfying the following?
(1) Point$(1,1)\in l_n$ for all $n\in \mathbb{Z}_{+}$.
(2) For all $n\in \mathbb{Z}_{+}$,$k_{n+1}=a_n-b_n$, where $k_{n+1}$ is the slope of $l_{n+1}$, $a_n,b_n$ are intercepts of $l_n$ on $x$-axis, $y$-axis.
(3) $k_nk_{n+1}\geq0$ for all $n\in \mathbb{Z}_{+}$.
1986 ITAMO, 4
Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates.
I tried to prove that any circle with center at $(0,0)$ has at most one point with coordinates $(a-\sqrt{2},b-\sqrt{3})$;$a,b \in \mathbb{Z}$.
So that when we translate the center to $(\sqrt{2},\sqrt{3})$ we have what we wanted to show.
PEN C Problems, 5
Let $p$ be an odd prime and let $Z_{p}$ denote (the field of) integers modulo $p$. How many elements are in the set \[\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?\]
2012 NIMO Summer Contest, 14
A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$.
[i]Proposed by Lewis Chen[/i]
1989 AMC 12/AHSME, 16
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 46$
1971 AMC 12/AHSME, 3
If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the xy-plane, then $x$ is equal to
$\textbf{(A) }-2\qquad\textbf{(B) }2\qquad\textbf{(C) }-8\qquad\textbf{(D) }6\qquad \textbf{(E) }-6$
2022 USAMTS Problems, 5
A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.
1992 AMC 12/AHSME, 4
If $m > 0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m = $
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \sqrt{5} $
2023 AMC 12/AHSME, 10
In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
$\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$
2012 NIMO Problems, 14
A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$.
[i]Proposed by Lewis Chen[/i]
2012 Iran MO (3rd Round), 1
Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.
2009 Today's Calculation Of Integral, 423
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
2011 NIMO Problems, 1
A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$?
[i]Proposed by Eugene Chen[/i]
2012 NIMO Problems, 1
In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?
[i]Proposed by Aaron Lin[/i]
2010 Contests, 3
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
[img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]
2008 Harvard-MIT Mathematics Tournament, 2
([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.
1956 AMC 12/AHSME, 21
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$
1961 AMC 12/AHSME, 3
If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:
${{ \textbf{(A)}\ \pm \frac{2}{3} \qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ -\frac{3}{2} \qquad\textbf{(D)}\ 6}\qquad\textbf{(E)}\ -6} $
2011 AIME Problems, 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.