Found problems: 475
2003 National High School Mathematics League, 3
Line passes the focal point $F$ of parabola $y^2=8(x+2)$ with bank angle of $60^{\circ}$ intersects the parabola at $A,B$. Perpendicular bisector of $AB$ intersects $x$-axis at $P$, then the length of $PF$ is
$\text{(A)}\frac{16}{3}\qquad\text{(B)}\frac{8}{3}\qquad\text{(C)}\frac{16}{3}\sqrt3\qquad\text{(D)}8\sqrt3$
1976 Euclid, 4
Source: 1976 Euclid Part A Problem 4
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The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$
2011 Romanian Master of Mathematics, 3
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
2022 IMO Shortlist, G6
Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.
1969 IMO Shortlist, 5
$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
2011 Today's Calculation Of Integral, 703
Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.
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$
1990 Vietnam National Olympiad, 1
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.
2008 Stars Of Mathematics, 3
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
[i]Dan Schwarz[/i]
2009 Today's Calculation Of Integral, 472
Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.
2007 ITest, 32
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer?
[asy]
import graph;
size(300);
defaultpen(linewidth(0.8)+fontsize(10));
real k=1.5;
real endp=sqrt(k);
real f(real x) {
return k-x^2;
}
path parabola=graph(f,-endp,endp)--cycle;
filldraw(parabola, lightgray);
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));
label("Region I", (0,2*k/5));
label("Box II", (51/64*endp,13/16*k));
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));
[/asy]
2014 Contests, 3
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
2011 Tournament of Towns, 4
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for
a) $N = 2011$
b) $N = 2012$ ?
2010 Today's Calculation Of Integral, 655
Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.
1969 IMO Longlists, 2
$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$
$(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$
$(c)$ Find the locus of the centers of these hyperbolas.
$(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$
1940 Putnam, A5
Prove that the simultaneous equations
$$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$
are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.
2005 Taiwan TST Round 2, 3
In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.
2013 Stanford Mathematics Tournament, 5
For exactly two real values of $b$, $b_1$ and $b_2$, the line $y=bx-17$ intersects the parabola $y=x^2 +2x+3$ at exactly one point. Compute $b_1^2+b_2^2$.
1998 IberoAmerican Olympiad For University Students, 2
In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$.
Find both the locus of the centroid of all such triangles and its area.
2007 Moldova Team Selection Test, 1
Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
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 $l_1,\ l_2$.
1999 Federal Competition For Advanced Students, Part 2, 3
Find all pairs $(x, y)$ of real numbers such that
\[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\]
where $f(x)=[x]$ is the floor function.
2007 Moldova National Olympiad, 12.6
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
2008 Harvard-MIT Mathematics Tournament, 6
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
1963 Putnam, A6
Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$, and let $AB$ and $CD$ be any two other chords through $M$. If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$, prove that $M$ is the midpoint of the segment $PQ.$