Found problems: 253
2011 China Second Round Olympiad, 7
The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$. Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$. Find the coordinate of point $C$.
2009 Today's Calculation Of Integral, 476
Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.
2020 Tournament Of Towns, 1
Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ?
Alexey Tolpygo
2013 Princeton University Math Competition, 3
The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?
2011 AIME Problems, 6
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2010 Contests, 2
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
2012 Today's Calculation Of Integral, 790
Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$.
(1) Find the equations of the tangents $l_1,\ l_2$.
(2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.
1969 AMC 12/AHSME, 26
[asy]
size(180);
defaultpen(linewidth(0.8));
real r=4/5;
draw((-1,0)..(-6/7,r/3)..(0,r)..(6/7,r/3)..(1,0),linetype("4 4"));
draw((-1,0)--(1,0)^^origin--(0,r));
label("$A$",(-1,0),W);
label("$B$",(1,0),E);
label("$M$",origin,S);
label("$C$",(0,r),N);
[/asy]
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:
$\textbf{(A) }1\qquad
\textbf{(B) }15\qquad
\textbf{(C) }15\tfrac13\qquad
\textbf{(D) }15\tfrac12\qquad
\textbf{(E) }15\tfrac34$
2012 Today's Calculation Of Integral, 809
For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$.
Find the maximum area of $S(a)$.
2014 HMNT, 8
Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying
$$(y + x) = (y - x)^2 + 3(y - x) + 3.$$
Find the minimum possible value of $y$.
1998 Harvard-MIT Mathematics Tournament, 7
A parabola is inscribed in equilateral triangle $ABC$ of side length $1$ in the sense that $AC$ and $BC$ are tangent to the parabola at $A$ and $B$, respectively.
Find the area between $AB$ and the parabola.
2025 AIME, 9
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.
2009 Today's Calculation Of Integral, 504
Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.
2007 ITest, 51
Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]
2010 Tournament Of Towns, 2
Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$
1980 Putnam, A1
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 $I_j .$
1946 Putnam, B2
Let $A, B$ be two variable points on a parabola $P_{0}$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$. If the equation of $P_0$ is $y=m x^2$, find the equation of $P_n .$
Kvant 2021, M2634
Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$.
[i]Proposed by M. Panov[/i]
2007 ITest, -1
The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
1998 All-Russian Olympiad, 1
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
2012 Belarus Team Selection Test, 2
Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above.
(I. Voronovich)
1957 AMC 12/AHSME, 10
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
Denmark (Mohr) - geometry, 1999.1
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.
2007 Today's Calculation Of Integral, 177
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$