Olympiad problems

2007 Turkey Team Selection Test, 1

[color=indigo]Let $ABC$ is an acute angled triangle and let $A_{1},\, B_{1},\, C_{1}$ are points respectively on $BC,\,CA,\,AB$ such that $\triangle ABC$ is similar to $\triangle A_{1}B_{1}C_{1}.$ Prove that orthocenter of $A_{1}B_{1}C_{1}$ coincides with circumcenter of $ABC$.[/color]

2011 Romanian Masters In Mathematics, 3

Tags: analytic geometry , modular arithmetic , combinatorics

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]

2011 India Regional Mathematical Olympiad, 5

Tags: geometry , analytic geometry , projective geometry

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively. If $AC,BD,EG,FH$ concur at a point $O,$ prove that $ABCD$ is a parallelogram.

2011 AMC 12/AHSME, 23

Tags: analytic geometry , geometry , rectangle

A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? $ \textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255 $

2012 Online Math Open Problems, 44

Tags: geometric transformation , reflection , analytic geometry , induction , pigeonhole principle , geometry

Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide. [i]Author: Anderson Wang[/i]

2005 IberoAmerican Olympiad For University Students, 1

Tags: polynomial , analytic geometry , limit , induction , ratio , algebra

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

2012 Math Prize For Girls Problems, 16

Tags: geometry , ellipse , analytic geometry , absolute value , conic

Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?

2006 Swedish Mathematical Competition, 3

Tags: analytic geometry , graphing lines , slope , calculus , derivative , algebra

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

V Soros Olympiad 1998 - 99 (Russia), 11.3

Tags: analytic geometry , area , geometry

Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.

1999 Hungary-Israel Binational, 2

Tags: analytic geometry , graphing lines , slope , combinatorics

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

2009 AMC 12/AHSME, 24

Tags: trigonometry , function , mathcamp , analytic geometry

For how many values of $ x$ in $ [0,\pi]$ is $ \sin^{\minus{}1}(\sin 6x)\equal{}\cos^{\minus{}1}(\cos x)$? Note: The functions $ \sin^{\minus{}1}\equal{}\arcsin$ and $ \cos^{\minus{}1}\equal{}\arccos$ denote inverse trigonometric functions. $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

1976 AMC 12/AHSME, 22

Tags: analytic geometry

Given an equilateral triangle with side of length $s$, consider the locus of all points $\mathit{P}$ in the plane of the triangle such that the sum of the squares of the distances from $\mathit{P}$ to the vertices of the triangle is a fixed number $a$. This locus $\textbf{(A) }\text{is a circle if }a>s^2\qquad$ $\textbf{(B) }\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad$ $\textbf{(C) }\text{is a circle with positive radius only if }s^2<a<2s^2\qquad$ $\textbf{(D) }\text{contains only a finite number of points for any value of }a\qquad $ $\textbf{(E) }\text{is none of these}$

2011 Today's Calculation Of Integral, 756

Tags: calculus , integration , geometry , analytic geometry , calculus computations

Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$ Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.

2021 BMT, 1

Tags: analytic geometry , geometry

The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine?

2007 Paraguay Mathematical Olympiad, 3

Tags: ratio , analytic geometry , graphing lines , slope

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

2003 France Team Selection Test, 1

Tags: analytic geometry , modular arithmetic , number theory

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

2007 Iran Team Selection Test, 2

Tags: analytic geometry , trigonometry , inequalities , circumcircle , geometry

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

1990 IMO Shortlist, 28

Tags: analytic geometry , vector , number theory , rational number

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

2005 AMC 12/AHSME, 12

Tags: analytic geometry , graphing lines , slope , greatest common divisor , geometry , rectangle , number theory

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

1989 National High School Mathematics League, 3

Tags: function , analytic geometry

For any function $f(x)$, in the same rectangular coordinates, figures of function $y=f(x-1)$ and $y=f(-x+1)$ $\text{(A)}$ are symmetrical about $x$-axis $\text{(B)}$ are symmetrical about line $x=1$ $\text{(C)}$ are symmetrical about line $x=-1$ $\text{(D)}$ are symmetrical about $y$-axis

1985 IMO Longlists, 97

Tags: analytic geometry , symmetry , trigonometry , function , geometry , perpendicular bisector , pythagorean theorem

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

2012 India National Olympiad, 4

Tags: ratio , geometry , analytic geometry , geometric transformation , homothety , calculus , integration

Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.

2004 National High School Mathematics League, 12

Tags: analytic geometry , geometry

In rectangular coordinate system, give two points $M(-1,2),N(1,4)$, $P$ is a moving point on $x$-axis, when $\angle MPN$ takes its maximum value, the $x$-axis of $P$ is________.

2001 AIME Problems, 9

Tags: ratio , analytic geometry , trigonometry , number theory , relatively prime , geometry

In triangle $ABC$, $AB=13,$ $BC=15$ and $CA=17.$ Point $D$ is on $\overline{AB},$ $E$ is on $\overline{BC},$ and $F$ is on $\overline{CA}.$ Let $AD=p\cdot AB,$ $BE=q\cdot BC,$ and $CF=r\cdot CA,$ where $p,$ $q,$ and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5.$ The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1981 AMC 12/AHSME, 22

Tags: analytic geometry , geometry , 3d geometry

How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four? $\text{(A)} \ 60 \qquad \text{(B)} \ 64 \qquad \text{(C)} \ 72 \qquad \text{(D)} \ 76 \qquad \text{(E)} \ 100$