Olympiad problems

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 $\lambda=$________.

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'$.

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PEN C Problems, 5

Let $p$ be an odd prime and let $Z_{p}$ denote (the field of) integers modulo $p$. How many elements are in the set \[\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?\]

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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.

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2006 Stanford Mathematics Tournament, 13

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

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2005 IberoAmerican Olympiad For University Students, 4

A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$. Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola. [b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.

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