Olympiad problems

2011 ELMO Shortlist, 4

Tags: modular arithmetic , conic , ellipse , geometric transformation , reflection , geometry

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$. [i]Calvin Deng.[/i]

1941 Putnam, B1

Tags: hyperbola , conic

A particle $(x,y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that $$y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy,$$ and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center.

1982 All Soviet Union Mathematical Olympiad, 339

Tags: geometric construction , parabola , geometry , conic

There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?

2018 CMIMC Geometry, 9

Tags: ellipse , geometry , conic

Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?

1986 National High School Mathematics League, 7

Tags: ellipse , conic

Inside a circular column with a bottom surface with radius of $6$, there are two balls with radius of $6$ as well. The distance betwen their centers is $13$. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve $C$. $C$ is a ellipse, then the sum of its short axis and long axis is________.

2009 Today's Calculation Of Integral, 419

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

In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant. (1) Find the equation of $ l$. (2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.

1979 IMO Longlists, 71

Tags: geometry , parallelogram , circles , fixed point , conic

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

2003 AMC 12-AHSME, 19

Tags: parabola , function , geometry , geometric transformation , reflection , analytic geometry , conic

A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reﬂected about the $ x$-axis. The parabola and its reﬂection are translated horizontally ﬁve units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

2008 Stars Of Mathematics, 3

Tags: geometry , incenter , geometric transformation , homothety , parallelogram , ellipse , conic

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

2015 IMAR Test, 4

Tags: function , interval , polynomial , analysis , real analysis , algebra , conic

(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$. (b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.

2007 Today's Calculation Of Integral, 211

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

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

2008 Harvard-MIT Mathematics Tournament, 12

Tags: geometry , 3d geometry , sphere , ellipse , ratio , geometric series , conic

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

2011 ITAMO, 4

Tags: ellipse , geometry , italy , conic

$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

2012 ELMO Problems, 5

Tags: geometry , parallelogram , trigonometry , ellipse , geometric transformation , reflection , conic

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

2007 Today's Calculation Of Integral, 193

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

For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$. Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin. (1) Find the equation of $l$. (2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.

2019 Belarusian National Olympiad, 11.1

Tags: parabola , hyperbola , geometry , circumcircle , conic

[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]

2008 Moldova National Olympiad, 12.7

Tags: hyperbola , geometry , conic

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Kvant 2025, M2831

Tags: geometry , parabola , conic

Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$). [i]From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov[/i]

1965 AMC 12/AHSME, 9

Tags: parabola , calculus , conic

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$

2011 ELMO Shortlist, 4

Tags: modular arithmetic , ellipse , geometric transformation , reflection , geometry , conic

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$. [i]Calvin Deng.[/i]

2006 AMC 12/AHSME, 21

Tags: ellipse , geometry , rectangle , perimeter , conic

Rectangle $ ABCD$ has area 2006. An ellipse with area $ 2006\pi$ passes through $ A$ and $ C$ and has foci at $ B$ and $ D$. What is the perimeter of the rectangle? (The area of an ellipse is $ \pi ab$, where $ 2a$ and $ 2b$ are the lengths of its axes.) $ \textbf{(A) } \frac {16\sqrt {2006}}{\pi} \qquad \textbf{(B) } \frac {1003}4 \qquad \textbf{(C) } 8\sqrt {1003} \qquad \textbf{(D) } 6\sqrt {2006} \qquad \textbf{(E) } \frac {32\sqrt {1003}}\pi$

2014 Contests, 2

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

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2022 Sharygin Geometry Olympiad, 23

Tags: ellipse , geometry , conic

An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

2016 ASMT, T3

Tags: ellipse , geometry , analytic geometry , conic

An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.

2013 Stanford Mathematics Tournament, 5

Tags: parabola , conic

For exactly two real values of $b$, $b_1$ and $b_2$, the line $y=bx-17$ intersects the parabola $y=x^2 +2x+3$ at exactly one point. Compute $b_1^2+b_2^2$.