Found problems: 487
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 $S$.
$O$ is the vertex of a parabola, $F$ is its focus. $PQ$ is a chord of the parabola. If $|OF|=a,|PQ|=b$, find the area of $\triangle OPQ$.
Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.
Let $a,\ b$ be real numbers. Suppose that a function $f(x)$ satisfies $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$ and has the maximum value $2\pi$ for $-\pi \leq x\leq \pi$.
Find the minimum value of $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$
[i]2010 Chiba University entrance exam[/i]
Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?
Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
(1) Find $a,\ b,\ c,\ d$.
(2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.
Let $ABC$ be a triangle, and let $D$, $E$, $F$ be 3 points on the sides $BC$, $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$, $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$. Prove that $D$, $E$, $F$ are the midpoints of the sides of the triangle $ABC$.
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.
Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]
Left focal point and right focal point of a hyperbola are $F_1,F_2$, left focal point and right focal point of a hyperbola are $M,N$. If $P$ is a point on the hyperbola, then the tangent point of inscribed circle of $\triangle PF_1F_2$ on $F_1F_2$ is
$\text{(A)}$a point on segment $MN$
$\text{(B)}$a point on segment $F_1M$ or $F_2N$
$\text{(C)}$point $M$ or $N$
$\text{(D)}$not sure
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.
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$.
The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$?
$ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$
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 $\exists k\in\mathbb{N}\cup\{0\}$, $\overline{OP_k}^2=2017$.
Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.
[i](Cyprus)[/i]
A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$.
[i]Evan O' Dorney.[/i]
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
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.
Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.
Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.
$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.
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 coeffi cient 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.)
Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.
Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?