Source: 1976 Euclid Part A Problem 4 ----- 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 $\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$

The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(200); real f(real x) {return 1.2*exp(2/3*log(16-x^2));} path Q=graph(f,-3.99999,3.99999); path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle}; for(int k=0;k<3;++k) { fill(P[k],grey); draw(P[k]); } draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]

The graph of $x^2-4y^2=0$ is: ${{ \textbf{(A)}\ \text{a parabola} \qquad\textbf{(B)}\ \text{an ellipse} \qquad\textbf{(C)}\ \text{a pair of straight lines} \qquad\textbf{(D)}\ \text{a point} }\qquad\textbf{(E)}\ \text{none of these} } $

Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]

[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: $\textbf{(A) }1\qquad \textbf{(B) }15\qquad \textbf{(C) }15\tfrac13\qquad \textbf{(D) }15\tfrac12\qquad \textbf{(E) }15\tfrac34$

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

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$

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.

On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of parabola $y^2=x$. Compute the sum of all possible areas of $ABCD$.

$A(1,1)$ is a point on parabola $y=x^2$. Draw the tangent line of the parabola that passes $A$, the line intersects $x$-axis at $D$, intersects $y$-axis at $B$. $C$ is a point on the parabola, and $E$ is a point on segment $AC$, such that $\frac{AE}{EC}=\lambda_1$, $F$ is a point on segment $BC$, such that $\frac{BF}{FC}=\lambda_2$. If $\lambda_1+\lambda_2=1$, $CD$ and $EF$ intersect at $P$. When $C$ moves, find the path equation of $P$.

If $ y\equal{}x^2\plus{}px\plus{}q$, then if the least possible value of $ y$ is zero $ q$ is equal to: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{p^2}{4} \qquad \textbf{(C)}\ \frac{p}{2} \qquad \textbf{(D)}\ \minus{}\frac{p}{2} \qquad \textbf{(E)}\ \frac{p^2}{4}\minus{}q$

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.

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is: $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \text{not finite}$

Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that \[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 \]The area of $ R$ is closest to $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 25$

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$. Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.

A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.

Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$