Olympiad problems

2011 Albania Team Selection Test, 3

Tags: ratio , hyperbola , trigonometry , analytic geometry , circumcircle , geometry , conic

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

2009 Spain Mathematical Olympiad, 6

Tags: symmetry , geometry , geometric transformation , reflection , trapezoid , ellipse , conic

Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.

2016 Israel Team Selection Test, 3

Tags: geometry , 3d geometry , octahedron , conic

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

2011 AIME Problems, 10

Tags: probability , parabola , conic

The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.

2017 BMT Spring, 19

Tags: ellipse , geometry , analytic geometry , conic

Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.

1950 AMC 12/AHSME, 49

Tags: ellipse , ratio , trigonometry , conic

A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is: $\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ \textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ \textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ \textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ \textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$

1987 China Team Selection Test, 1

Tags: geometry , rectangle , ratio , analytic geometry , symmetry , hyperbola , conic

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

2006 AMC 10, 8

Tags: quadratic , parabola , conic

A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$? $ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$

1941 Putnam, A7

Tags: linear algebra , matrix , ellipse , conic

Do either (1) or (2): (1) Prove that the determinant of the matrix $$\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix}$$ is given by $(1+a^2 +b^2 +c^2)^{3}$. (2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.

2012 ISI Entrance Examination, 7

Tags: ellipse , geometry , conic

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$

1952 AMC 12/AHSME, 13

Tags: quadratic , function , parabola , calculus , conic

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

2024 Sharygin Geometry Olympiad, 22

Tags: ellipse , geometry , perpendicular bisector , conic

A segment $AB$ is given. Let $C$ be an arbitrary point of the perpendicular bisector to $AB$; $O$ be the point on the circumcircle of $ABC$ opposite to $C$; and an ellipse centred at $O$ touch $AB, BC, CA$. Find the locus of touching points of the ellipse with the line $BC$.

2013 Today's Calculation Of Integral, 874

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2001 National High School Mathematics League, 7

Tags: ellipse , conic

The length of minor axis of ellipse $\rho-\frac{1}{2-\cos\theta}$ is________.

2010 CHMMC Winter, 1

Tags: conic , algebra , polynomial

The monic polynomial $f$ has rational coefficients and is irreducible over the rational numbers. If $f(\sqrt5 +\sqrt2)= 0$, compute $f(f(\sqrt5 -\sqrt2))$. (A polynomial is [i]monic [/i] if its leading coefficient is $1$. A polynomial is [i]irreducible [/i] over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, $x^2 - 2$ is irreducible, but $x^2 - 1 = (x + 1)(x - 1)$ is not.)

Revenge EL(S)MO 2024, 2

Tags: ellipse , conic , geometry

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by [i]Benny Wang + Oron Wang[/i]

1999 Denmark MO - Mohr Contest, 1

Tags: parabola , tangent , conic , circles

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

1987 IberoAmerican, 2

Tags: geometry , geometric transformation , reflection , hyperbola , symmetry , trigonometry , conic

In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.

2011 Today's Calculation Of Integral, 675

Tags: calculus , integration , analytic geometry , parabola , trapezoid , conic , geometry

In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$. [i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]

2013 ISI Entrance Examination, 8

Tags: parabola , analytic geometry , conic

Let $ABCD$ be a square such that $AB$ lies along the line $y=x+8,$ and $C$ and $D$ lie on the parabola $y=x^2.$ Find all possible values of sidelength of the square.

1992 AMC 12/AHSME, 14

Tags: conic

Which of the following equations have the same graph? $I.\ y = x - 2$ $II.\ y = \frac{x^{2} - 4}{x + 2}$ $III.\ (x + 2)y = x^{2} - 4$ $ \textbf{(A)}\ \text{I and II only} $ $ \textbf{(B)}\ \text{I and III only} $ $ \textbf{(C)}\ \text{II and III only} $ $ \textbf{(D)}\ \text{I, II and III} $ $ \textbf{(E)}\ \text{None. All equations have different graphs} $

2018 Belarusian National Olympiad, 11.5

Tags: algebra , hyperbola , conic

The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.

1979 Miklós Schweitzer, 6

Tags: analytic geometry , conic , advanced fields

Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $ \mathbb{E}^3$ not lying on the $ z$-axis by the metric tensor \[ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \minus{}\sqrt{x^2\plus{}y^2} \\ \end{array} \right),\] where $ (x,y,z)$ is a Cartesian coordinate system $ \mathbb{E}^3$. Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $ (x,y)$-plane are straight lines or conic sections with focus at the origin [i]P. Nagy[/i]

2021 Simon Marais Mathematical Competition, A1

Tags: parabola , calculus , conic

Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation \[ y = ax^2 + bx + c, \] and the lines defined by the six equations \begin{align*} &y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\ &y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.

1969 IMO Shortlist, 4

Tags: geometry , locus , conic

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$