Olympiad problems

2008 Moldova National Olympiad, 12.3

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

2020 Bangladesh Mathematical Olympiad National, Problem 4

Tags: geometry , analytic geometry , graphing lines , slope

$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?

1969 AMC 12/AHSME, 18

Tags: analytic geometry , graphing lines , slope

The number of points common to the graphs of \[(x-y+2)(3x+y-4)=0\text{ and }(x+y-2)(2x-5y+7)=0\] is: $\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{infinite}$

1991 Arnold's Trivium, 1

Tags: calculus , derivative , integration , function , analytic geometry , graphing lines , slope

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

2009 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , function , analytic geometry , graphing lines , slope

Let $f$ be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?

2003 AMC 12-AHSME, 24

Tags: ceiling function , analytic geometry , graphing lines , slope , function , algebra , system of equations

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

1950 AMC 12/AHSME, 14

Tags: analytic geometry , graphing lines , slope

For the simultaneous equations \[ 2x\minus{}3y\equal{}8\] \[ 6y\minus{}4x\equal{}9\] $\textbf{(A)}\ x=4,y=0 \qquad \textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad \textbf{(C)}\ x=0,y=0 \qquad\\ \textbf{(D)}\ \text{There is no solution} \qquad \textbf{(E)}\ \text{There are an infinite number of solutions}$

2013 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , floor function , ceiling function , analytic geometry , graphing lines , slope

Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]

1957 AMC 12/AHSME, 34

Tags: inequalities , parameterization , analytic geometry , graphing lines , slope , algebra , function

The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

2012 NIMO Problems, 14

Tags: analytic geometry , graphing lines , slope

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]

2006 AMC 10, 20

Tags: geometry , rectangle , analytic geometry , graphing lines , slope , vector

In rectangle $ ABCD$, we have $ A \equal{} (6, \minus{} 22)$, $ B \equal{} (2006,178)$, and $ D \equal{} (8,y)$, for some integer $ y$. What is the area of rectangle $ ABCD$? $ \textbf{(A) } 4000 \qquad \textbf{(B) } 4040 \qquad \textbf{(C) } 4400 \qquad \textbf{(D) } 40,000 \qquad \textbf{(E) } 40,400$

1986 AMC 12/AHSME, 26

Tags: analytic geometry , graphing lines , slope

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $

1984 AMC 12/AHSME, 29

Tags: calculus , quadratic , trigonometry , analytic geometry , graphing lines , slope , algebra

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

2020 AMC 12/AHSME, 7

Tags: analytic geometry , graphing lines , slope

Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

1988 National High School Mathematics League, 3

Tags: analytic geometry , graphing lines , slope

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 $n\in \mathbb{Z}_{+}$.

2007 Iran Team Selection Test, 3

Tags: analytic geometry , graphing lines , slope , trigonometry , geometry , geometric transformation , reflection

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

2005 Taiwan National Olympiad, 3

Tags: ellipse , analytic geometry , graphing lines , slope , geometric transformation , conic , geometry

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

1963 AMC 12/AHSME, 7

Tags: analytic geometry , graphing lines , slope

Given the four equations: $\textbf{(1)}\ 3y-2x=12 \qquad \textbf{(2)}\ -2x-3y=10 \qquad \textbf{(3)}\ 3y+2x=12 \qquad \textbf{(4)}\ 2y+3x=10$ The pair representing the perpendicular lines is: $\textbf{(A)}\ \text{(1) and (4)} \qquad \textbf{(B)}\ \text{(1) and (3)} \qquad \textbf{(C)}\ \text{(1) and (2)} \qquad \textbf{(D)}\ \text{(2) and (4)} \qquad \textbf{(E)}\ \text{(2) and (3)}$

2013 NIMO Problems, 2

Tags: geometry , perimeter , probability , analytic geometry , graphing lines , slope , geometric transformation

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

2007 F = Ma, 26

Tags: analytic geometry , graphing lines , slope , trigonometry , rotation

A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? $ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $

2012 NIMO Problems, 1

Tags: analytic geometry , graphing lines , slope

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]

2007 Putnam, 1

Tags: symmetry , analytic geometry , graphing lines , slope , quadratic , geometry

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

2000 Cono Sur Olympiad, 1

Tags: circumcircle , analytic geometry , graphing lines , slope , geometry

In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.

2010 Contests, 3

Tags: geometry , rectangle , parallelogram , analytic geometry , graphing lines , slope , combinatorics

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 Balkan MO Shortlist, C3

Tags: geometry , rectangle , analytic geometry , graphing lines , slope , geometric transformation , reflection

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.