Olympiad problems

2015 Sharygin Geometry Olympiad, P20

Tags: conics , ellipse , Locus , geometry , circumcircle

Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.

2012 All-Russian Olympiad, 3

Tags: conics , parabola , geometry , quadratics , analytic geometry , combinatorics unsolved , combinatorics

On a Cartesian plane, $n$ parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than $2(n-1)$ corners (i.e. the intersections of a pair of parabolas).

2013 Today's Calculation Of Integral, 895

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

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

2013 Today's Calculation Of Integral, 891

Tags: calculus , integration , geometry , 3D geometry , conics , hyperbola , rotation

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

1996 VJIMC, Problem 1

Tags: conics , parabola , geometry

Is it possible to cover the plane with the interiors of a finite number of parabolas?

2008 IMS, 3

Tags: geometry , conics , parabola , ratio , function , calculus , derivative

Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.

2009 Spain Mathematical Olympiad, 6

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

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.

2004 Brazil National Olympiad, 1

Tags: geometry , rhombus , conics , hyperbola , symmetry , inradius , incenter

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

2008 Hungary-Israel Binational, 3

Tags: geometry , conics , hyperbola , geometric transformation , reflection , angle bisector , geometry unsolved

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

2001 AMC 12/AHSME, 13

Tags: conics , parabola , geometry , geometric transformation , reflection , function , AMC

The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$? $ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$

2013 Princeton University Math Competition, 5

Tags: AMC , USA(J)MO , USAMO , conics , parabola

Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.

2005 National High School Mathematics League, 11

Tags: geometry , conics , parabola

One side of a square in on line $y=2x-17$, and two other points are on parabola $y=x^2$, then the minumum value of the area of the square is________.

1962 AMC 12/AHSME, 19

Tags: conics , parabola

If the parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ passes through the points $ ( \minus{} 1, 12), (0, 5),$ and $ (2, \minus{} 3),$ the value of $ a \plus{} b \plus{} c$ is: $ \textbf{(A)}\ \minus{} 4 \qquad \textbf{(B)}\ \minus{} 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Estonia Open Junior - geometry, 2016.2.4

Tags: conics , parabola , area , Quadratic , algebra , analytic geometry

Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

1977 AMC 12/AHSME, 5

Tags: geometry , conics , parabola , inequalities , perpendicular bisector , triangle inequality , AMC

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

2010 Putnam, B2

Tags: Putnam , analytic geometry , inequalities , geometry , rectangle , conics , hyperbola

Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$

2011 China Girls Math Olympiad, 3

Tags: inequalities , function , calculus , conics , hyperbola , China

The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.

2013 AMC 12/AHSME, 20

Tags: trigonometry , geometry , trapezoid , conics , parabola , analytic geometry , graphing lines

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

2019 Belarusian National Olympiad, 11.1

Tags: conics , parabola , hyperbola , geometry , circumcircle

[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points. [b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values. [i](I. Gorodnin)[/i]

1993 AMC 12/AHSME, 26

Tags: function , algebra , domain , inequalities , conics , parabola , calculus

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

2012 Today's Calculation Of Integral, 779

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

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

2008 Paraguay Mathematical Olympiad, 2

Tags: conics , parabola , absolute value

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$

2017 Taiwan TST Round 3, 2

Tags: conics , ellipse , number theory

Choose a rational point $P_0(x_p,y_p)$ arbitrary on ellipse $C:x^2+2y^2=2098$. Define $P_1,P_2,\cdots$ recursively by the following rules: $(1)$ Choose a lattice point $Q_i=(x_i,y_i)\notin C$ such that $|x_i|<50$ and $|y_i|<50$. $(2)$ Line $P_iQ_i$ intersects $C$ at another point $P_{i+1}$. Prove that for any point $P_0$ we can choose suitable points $Q_0,Q_1,\cdots$ such that $\exists k\in\mathbb{N}\cup\{0\}$, $\overline{OP_k}^2=2017$.

2010 Today's Calculation Of Integral, 569

Tags: calculus , integration , analytic geometry , geometry , graphing lines , slope , conics

In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\ 2)$ with the slope $ a$ and the parabola $ y\equal{}x^2$. When $ a$ varies in the range of $ 0\leq a\leq 6$, find the value of $ a$ such that $ S(a)$ is minimized.

2018 Belarusian National Olympiad, 10.6

Tags: analytic geometry , conics , parabola , geometry , circumcircle

The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$.