Olympiad problems

1952 Putnam, B6

Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.

Today's calculation of integrals, 854

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

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

1984 National High School Mathematics League, 3

Tags: analytic geometry , hyperbola , conic

For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent? Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$. $\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$

1962 AMC 12/AHSME, 19

Tags: parabola , conic

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$

1990 IberoAmerican, 4

Tags: parabola , rectangle , power of a point , radical axis , geometry , conic

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

1986 AMC 12/AHSME, 18

Tags: geometry , ellipse , conic

A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$

2014 District Olympiad, 1

Tags: ellipse , complex number , conic , algebra

Solve for $z\in \mathbb{C}$ the equation : \[ |z-|z+1||=|z+|z-1|| \]

2013 Romania Team Selection Test, 2

Tags: euler , geometry , circumcircle , geometric transformation , homothety , conic

The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle. EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]

2011 Today's Calculation Of Integral, 749

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

Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.

1942 Putnam, B2

Tags: parabola , conic

For the family of parabolas $$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$ (i) find the locus of vertices, (ii) find the envelope, (iii) sketch the envelope and two typical curves of the family.

2012 Today's Calculation Of Integral, 777

Tags: calculus , integration , parabola , geometry , analytic geometry , conic , graphing lines

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

2008 AMC 10, 19

Tags: geometry , 3d geometry , prism , ellipse , congruent triangles , conic

A cylindrical tank with radius $ 4$ feet and height $ 9$ feet is lying on its side. The tank is filled with water to a depth of $ 2$ feet. What is the volume of the water, in cubic feet? $ \textbf{(A)}\ 24\pi \minus{} 36 \sqrt {2} \qquad \textbf{(B)}\ 24\pi \minus{} 24 \sqrt {3} \qquad \textbf{(C)}\ 36\pi \minus{} 36 \sqrt {3} \qquad \textbf{(D)}\ 36\pi \minus{} 24 \sqrt {2} \\ \textbf{(E)}\ 48\pi \minus{} 36 \sqrt {3}$

2010 Today's Calculation Of Integral, 612

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

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

2005 International Zhautykov Olympiad, 2

Tags: geometry , perimeter , analytic geometry , linear algebra , parallelogram , conic

Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.

2022 JHMT HS, 6

Tags: parabola , geometry , conic

Triangle $ABC$ has side lengths $AC = 3$, $BC = 4$, and $AB = 5$. Let $I$ be the incenter of $\triangle{ABC}$, and let $\mathcal{P}$ be the parabola with focus $I$ and directrix $\overleftrightarrow{AC}$. Suppose that $\mathcal{P}$ intersects $\overline{AB}$ and $\overline{BC}$ at points $D$ and $E$, respectively. Find $DI+EI$.

1963 AMC 12/AHSME, 29

Tags: parabola , conic

A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: $\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$

1996 National High School Mathematics League, 1

Tags: ellipse , conic

Connect the commom points of circle$x^2+(y-1)^2=1$ and ellipse $9x^2+(y+1)^2=9$ with line segments, the figure is a $\text{(A)}$ line segment $\text{(B)}$ scalene triangle $\text{(C)}$ equilateral triangle $\text{(D)}$ quadrilateral

1980 Putnam, A1

Tags: parabola , algebra , polynomial , function , conic

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

1998 National High School Mathematics League, 15

Tags: parabola , analytic geometry , conic , geometry

Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$. Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.

2014 Belarus Team Selection Test, 1

Tags: geometry , analytic geometry , hyperbola , parallel , parabola , conic

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

2024 All-Russian Olympiad Regional Round, 11.7

Tags: quadratic , derivative , tangent , parabola , algebra , conic

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

2000 Irish Math Olympiad, 5

Tags: parabola , analytic geometry , linear algebra , matrix , algebra , conic

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

2006 China Second Round Olympiad, 13

Tags: parabola , conic

Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.

1997 Spain Mathematical Olympiad, 3

Tags: parabola , geometry , circumcircle , triangle , conic

For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.

2009 India Regional Mathematical Olympiad, 5

Tags: inequalities , ellipse , geometry , circumcircle , induction , function , conic

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.