Olympiad problems

2014 Harvard-MIT Mathematics Tournament, 7

Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

2000 Romania Team Selection Test, 3

Tags: geometry , 3d geometry , sphere , function , calculus , topology , analytic geometry

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

2022 Thailand Mathematical Olympiad, 5

Tags: function , functional equation , geometry , analytic geometry

Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation $$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$ for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.

1993 National High School Mathematics League, 4

Tags: analytic geometry , geometry

$C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$. The length of string of $C$ cut by $l:x=\frac{\pi}{4}$ is $d$. When $a$ changes, the minumum value of $d$ is $\text{(A)}\frac{\pi}{4}\qquad\text{(B)}\frac{\pi}{3}\qquad\text{(C)}\frac{\pi}{2}\qquad\text{(D)}\pi$

VII Soros Olympiad 2000 - 01, 9.1

Tags: analytic geometry , algebra

Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.

2006 Victor Vâlcovici, 2

Tags: geometry , circumcircle , analytic geometry

Consider a point $ B $ on a segment $ AC. $ Find the locus of the points $ M $ that have the property that the circumcircles of $ ABM $ and $ BCM $ have equal radii. [i]Nicolae Soare[/i]

2008 AMC 12/AHSME, 24

Tags: analytic geometry , quadratic

Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 21$

1992 Irish Math Olympiad, 5

Tags: geometry , analytic geometry

Let $ABC$ be a triangle such that the coordinates of the points $A$ and $B$ are rational numbers. Prove that the coordinates of $C$ are rational if, and only if, $\tan A$, $\tan B$, and $\tan C$, when defined, are all rational numbers.

2002 All-Russian Olympiad Regional Round, 10.5

Tags: parabola , analytic geometry , conic

Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.

2012 Iran MO (3rd Round), 3

Tags: ellipse , ratio , analytic geometry , geometry , geometric transformation , dilation , conic

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] [i]Proposed by Morteza Saghafian[/i]

2007 Nicolae Coculescu, 4

Tags: locus , analytic geometry , infimum , geometry

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

1951 AMC 12/AHSME, 33

Tags: analytic geometry , graphing lines , slope

The roots of the equation $ x^2 \minus{} 2x \equal{} 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair: $ \textbf{(A)}\ y \equal{} x^2, y \equal{} 2x \qquad\textbf{(B)}\ y \equal{} x^2 \minus{} 2x, y \equal{} 0 \qquad\textbf{(C)}\ y \equal{} x, y \equal{} x \minus{} 2$ $ \textbf{(D)}\ y \equal{} x^2 \minus{} 2x \plus{} 1, y \equal{} 1 \qquad\textbf{(E)}\ y \equal{} x^2 \minus{} 1, y \equal{} 2x \minus{} 1$ [i][Note: Abscissas means x-coordinate.][/i]

2008 Harvard-MIT Mathematics Tournament, 6

Tags: quadratic , inequalities , function , parabola , analytic geometry , absolute value , conic

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

1988 China Team Selection Test, 3

Tags: geometry , geometric transformation , vector , analytic geometry , combinatorics , lattice

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

JBMO Geometry Collection, 2013

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

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

2008 Romania Team Selection Test, 3

Tags: inequalities , vector , trigonometry , analytic geometry , geometry

Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side. [i]Note[/i]. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.

2003 China Team Selection Test, 2

Tags: ratio , inequalities , analytic geometry , linear algebra , matrix , geometry

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

2009 JBMO Shortlist, 1

Tags: ceiling function , analytic geometry , geometry , circumcircle , combinatorics

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

1983 Spain Mathematical Olympiad, 5

Tags: analytic geometry , square

Find the coordinates of the vertices of a square $ABCD$, knowing that $A$ is on the line $y -2x -6 = 0$, $C$ at $x = 0$ and $B$ is the point $(a, 0)$ , being $a = \log_{2/3}(16/81)$.

2007 Iran MO (2nd Round), 3

Tags: analytic geometry , combinatorics

In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)

2008 IMC, 4

Tags: inequalities , linear algebra , matrix , group theory , abstract algebra , analytic geometry , geometry

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

2013 China Team Selection Test, 3

Tags: analytic geometry , geometry , combinatorics

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.