Found problems: 487
1998 AMC 12/AHSME, 14
A parabola has vertex at $(4,-5)$ and has two $x$-intercepts, one positive and one negative. If this parabola is the graph of $y = ax^2 + bx + c$, which of $a$, $b$, and $c$ must be positive?
$ \textbf{(A)}\ \text{Only }a\qquad \textbf{(B)}\ \text{Only }b\qquad \textbf{(C)}\ \text{Only }c\qquad \textbf{(D)}\ \text{Only }a\text{ and }b\qquad \textbf{(E)}\ \text{None}$
Today's calculation of integrals, 870
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$
(1) Find all points of intersection of $E$ and $H$.
(2) Find the area of the region expressed by the system of inequality
\[\left\{
\begin{array}{ll}
3x^2+y^2\leq 3 &\quad \\
xy\geq \frac 34 , &\quad
\end{array}
\right.\]
1990 National High School Mathematics League, 3
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
2014 Bulgaria National Olympiad, 3
Let $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$.
Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$.
[i]Proposed by N. Beluhov[/i]
2017 Mathematical Talent Reward Programme, SAQ: P 1
A monic polynomial is a polynomial whose highest degree coefficient is 1. Let $P(x)$ and $Q(x)$ be monic polynomial with real coefficients and $degP(x)=degQ(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions then $P(x+1)=Q(x-1)$ has a real solution
2008 AIME Problems, 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2006 China Second Round Olympiad, 1
An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.
Today's calculation of integrals, 873
Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$
(1) Find the condition for which $C_1$ is inscribed in $C_2$.
(2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$.
(3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$.
60 point
2010 AMC 12/AHSME, 19
Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$?
$ \textbf{(A)}\ 45 \qquad
\textbf{(B)}\ 63 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 201 \qquad
\textbf{(E)}\ 1005$
2004 Brazil National Olympiad, 1
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.
2003 Mediterranean Mathematics Olympiad, 2
In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
2015 AMC 10, 24
For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?
$\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
2019 AMC 10, 10
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
1969 IMO Shortlist, 1
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
1998 Tuymaada Olympiad, 5
A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.
2012 Math Prize For Girls Problems, 16
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
2010 Today's Calculation Of Integral, 620
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]
Kvant 2025, M2831
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]
2022 JHMT HS, 6
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 $DI+EI$.
1964 AMC 12/AHSME, 24
Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum?
$ \textbf{(A)}\ \frac{a+b}{2} \qquad\textbf{(B)}\ a+b \qquad\textbf{(C)}\ \sqrt{ab} \qquad\textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad\textbf{(E)}\ \frac{a+b}{2ab} $
2005 Iran MO (3rd Round), 5
Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.
2012 Hitotsubashi University Entrance Examination, 3
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.
1982 All Soviet Union Mathematical Olympiad, 339
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?
2000 Harvard-MIT Mathematics Tournament, 7
Let $ABC$ be a triangle inscribed in the ellipse $\frac{x^2}{4} +\frac{y^2}{9}= 1$. If its centroid is the origin $(0,0)$, find its area.
2019 Saudi Arabia Pre-TST + Training Tests, 3.1
Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P(2x^2 - 4x)$ and $P(4x - 2x^2)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x),B(x)$ such that $A(x)B(x) = P(x)$ and $A(x) = B(x)$ has no root in $(-1.1)$