Found problems: 487
2008 Paraguay Mathematical Olympiad, 2
Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value:
$S = |n-1| + |n-2| + \ldots + |n-100|$
III Soros Olympiad 1996 - 97 (Russia), 9.2
It is known that the graph of a quadratic trinomial $y = x^2 + px + q$ touches the graph of a straight line $y = 2x + p$. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.
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
1979 IMO Shortlist, 22
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 $P.$
2011 Today's Calculation Of Integral, 689
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny
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:
$\textbf{(A) }1\qquad
\textbf{(B) }15\qquad
\textbf{(C) }15\tfrac13\qquad
\textbf{(D) }15\tfrac12\qquad
\textbf{(E) }15\tfrac34$
2008 Harvard-MIT Mathematics Tournament, 9
Let $ S$ be the set of points $ (a,b)$ with $ 0\le a,b\le1$ such that the equation \[x^4 \plus{} ax^3 \minus{} bx^2 \plus{} ax \plus{} 1 \equal{} 0\] has at least one real root. Determine the area of the graph of $ S$.
2005 China National Olympiad, 2
A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.
2012 Today's Calculation Of Integral, 779
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane.
When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.
1996 China National Olympiad, 1
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
2013 AMC 12/AHSME, 20
For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?
$ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $
1991 National High School Mathematics League, 14
$O$ 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 $\triangle OPQ$.
2002 IMC, 1
A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis.
Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.
1998 National High School Mathematics League, 11
If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.
2002 Czech and Slovak Olympiad III A, 5
A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)
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 $y$.
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} $
2007 Putnam, 2
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
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
1998 Tuymaada Olympiad, 5
A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.
2013 Waseda University Entrance Examination, 1
Given a parabola $C: y^2=4px\ (p>0)$ with focus $F(p,\ 0)$. Let two lines $l_1,\ l_2$ passing through $F$ intersect orthogonaly each other,
$C$ intersects with $l_1$ at two points $P_1,\ P_2$ and $C$ intersects with $l_2$ at two points $Q_1,\ Q_2$. Answer the following questions.
(1) Set the equation of $l_1$ as $x=ay+p$ and let the coordinates of $P_1,\ P_2$ as $(x_1,\ y_1),\ (x_2,\ y_2)$, respectively. Express $y_1+y_2,\ y_1y_2$ in terms of $a,\ p$.
(2) Show that $\frac{1}{P_1P_2}+\frac{1}{Q_1Q_2}$ is constant regardless of way of taking $l_1,\ l_2$.
2006 Iran Team Selection Test, 3
Let $l,m$ be two parallel lines in the plane.
Let $P$ be a fixed point between them.
Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$.
(By angle $EPF$ we mean the directed angle)
Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.
1984 AMC 12/AHSME, 22
Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is
A. a straight line
B. a parabola
C. part, but not all, of a parabola
D. one branch of a hyperbola
E. None of these
2013 Miklós Schweitzer, 11
[list]
(a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.
(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.
[/list]
[i]Proposed by Tran Quoc Binh[/i]
1997 National High School Mathematics League, 8
Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.