Olympiad problems

2013 AMC 10, 22

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

1981 AMC 12/AHSME, 2

Tags: geometry , pythagorean theorem

Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is $\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$

2008 ITest, 62

Tags: geometry , function , trigonometry , pythagorean theorem

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.

1973 AMC 12/AHSME, 1

Tags: geometry , perpendicular bisector , pythagorean theorem

A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length $ \textbf{(A)}\ 3\sqrt3 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 6\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ \text{ none of these}$

2001 AMC 10, 24

Tags: geometry , trapezoid , pythagorean theorem

In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 12.25 \qquad \textbf{(C)}\ 12.5 \qquad \textbf{(D)}\ 12.75 \qquad \textbf{(E)}\ 13$

2011 Purple Comet Problems, 14

Tags: trigonometry , pythagorean theorem , geometry

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

1965 AMC 12/AHSME, 16

Tags: geometry , ratio , analytic geometry , pythagorean theorem , area of a triangle , heron's formula

Let line $ AC$ be perpendicular to line $ CE$. Connect $ A$ to $ D$, the midpoint of $ CE$, and connect $ E$ to $ B$, the midpoint of $ AC$. If $ AD$ and $ EB$ intersect in point $ F$, and $ \overline{BC} \equal{} \overline{CD} \equal{} 15$ inches, then the area of triangle $ DFE$, in square inches, is: $ \textbf{(A)}\ 50 \qquad \textbf{(B)}\ 50\sqrt {2} \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ \frac {15}{2}\sqrt {105} \qquad \textbf{(E)}\ 100$

2014 NIMO Problems, 5

Tags: circumcircle , ratio , trigonometry , complex number , pythagorean theorem , geometry

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

2014 Online Math Open Problems, 14

Tags: incenter , inradius , pythagorean theorem , congruent triangles , geometry

Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$. [i]Proposed by Evan Chen[/i]

2011 Middle European Mathematical Olympiad, 6

Tags: circumcircle , trigonometry , pythagorean theorem , trig identities , law of cosines , geometry

Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.

1951 AMC 12/AHSME, 49

Tags: pythagorean theorem , geometry

The medians of a right triangle which are drawn from the vertices of the acute angles are $ 5$ and $ \sqrt {40}$. The value of the hypotenuse is: $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 2\sqrt {40} \qquad\textbf{(C)}\ \sqrt {13} \qquad\textbf{(D)}\ 2\sqrt {13} \qquad\textbf{(E)}\ \text{none of these}$

1961 AMC 12/AHSME, 38

Tags: inequalities , pythagorean theorem , geometry

Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$: $ \textbf{(A)}\ s^2\le8r^2$ $\qquad\textbf{(B)}\ s^2=8r^2$ $\qquad\textbf{(C)}\ s^2 \ge 8r^2$ ${\qquad\textbf{(D)}\ s^2\le4r^2 }$ ${\qquad\textbf{(E)}\ x^2=4r^2 } $

2011 International Zhautykov Olympiad, 1

Tags: geometry , trapezoid , circumcircle , parallelogram , rectangle , perpendicular bisector , pythagorean theorem

Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?

2010 AMC 10, 16

Tags: geometry , symmetry , pythagorean theorem

A square of side length $ 1$ and a circle of radius $ \sqrt3/3$ share the same center. What is the area inside the circle, but outside the square? $ \textbf{(A)}\ \frac{\pi}3 \minus{} 1 \qquad\textbf{(B)}\ \frac{2\pi}{9} \minus{} \frac{\sqrt3}3 \qquad\textbf{(C)}\ \frac{\pi}{18} \qquad\textbf{(D)}\ \frac14 \qquad\textbf{(E)}\ 2\pi/9$

1966 AMC 12/AHSME, 6

Tags: trigonometry , trig identities , law of cosines , geometry , law of sines , pythagorean theorem

$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, the length of chord $AC$, expressed in inches, is: $\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}$

2013 AIME Problems, 13

Tags: geometry , ratio , pythagorean theorem , area of a triangle , heron's formula , algebra , system of equations

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.