Olympiad problems

2010 USAJMO, 3

Tags: geometry , rectangle

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

2012-2013 SDML (Middle School), 12

Tags: geometry , 3d geometry

For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

Ukrainian TYM Qualifying - geometry, 2015.20

Tags: ratio , inradii , geometric inequality , geometry

What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?

2008 Vietnam Team Selection Test, 1

Tags: geometric transformation , reflection , ratio , angle bisector , perpendicular bisector , geometry

On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$. $ 1.$ Prove that $ K$ always lie on a fixed line. $ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.

2019 ELMO Shortlist, G3

Tags: geometry

Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$. [i]Proposed by Ankit Bisain[/i]

2019 LIMIT Category A, Problem 2

Tags: geometry

Let $ABCD$ be a quadrilateral with sides $\left|\overline{AB}\right|=2$, $\left|\overline{BC}\right|=\left|\overline{CD}\right|=4$ and $\left|\overline{DA}\right|=5$. The opposite angles, $\angle A$ and $\angle C$ are equal. The length of diagonal $BD$ equals $\textbf{(A)}~2\sqrt6$ $\textbf{(B)}~3\sqrt3$ $\textbf{(C)}~3\sqrt6$ $\textbf{(D)}~2\sqrt3$

2012 Iran MO (3rd Round), 3

Tags: induction , pigeonhole principle , geometry , perpendicular bisector , algebra , binomial theorem , combinatorics

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

2018 AMC 12/AHSME, 8

Tags: geometry

Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$

1985 Spain Mathematical Olympiad, 3

Tags: trigonometric equation , trigonometry , geometry

Solve the equation $tan^2 2x+2 tan2x tan3x = 1$

2023 Moldova Team Selection Test, 10

Tags: geometry

Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$

2008 IMO Shortlist, 7

Tags: geometry , incenter , convex quadrilateral

Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$. [i]Author: Vladimir Shmarov, Russia[/i]

1965 AMC 12/AHSME, 4

Tags: geometry , geometric transformation , reflection

Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

1969 IMO Longlists, 51

Tags: geometry , calculus , counting , function

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

2016 Japan MO Preliminary, 4

Tags: combinatorics , rectangle , geometry

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

2015 Azerbaijan IMO TST, 1

Tags: geometry , circumcircle

Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$

2018 Mediterranean Mathematics OIympiad, 2

Tags: geometry , circumcircle , ugly

Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that $$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$

2008 Hanoi Open Mathematics Competitions, 8

Tags: geometry

Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$; $M, N$ be the centroid of $\Delta AOB$ and $\Delta COD$ and $P, Q$ be orthocenter of $\Delta BOC$ and $\Delta DOA$, respectively. Prove that $MN\bot PQ$.

1999 Tournament Of Towns, 2

Tags: ratio , square , geometry , right triangle

$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$. (A Blinkov)

2012 IFYM, Sozopol, 7

Tags: geometry , rotation , quadrilateral , parallelogram

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

2023 AMC 12/AHSME, 3

Tags: geometry

A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$? $\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$

2017 Argentina National Math Olympiad Level 2, 5

Tags: equal segments , angle bisector , circumradius , geometry

Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$. Note: The circumcircle of a triangle is the circle that passes through its three vertices.

1984 AMC 12/AHSME, 18

Tags: analytic geometry , geometry , inradius , incenter

A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is A. $\sqrt{2} - 1$ B. $\frac{1}{2}$ C. $2 - \sqrt{2}$ D. 1 E. Not uniquely determined

1994 Vietnam National Olympiad, 2

Tags: 3d geometry , sphere , tetrahedron , vector , analytic geometry , geometry

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

2019 Iran MO (3rd Round), 3

Tags: circumcircle , incenter , geometry

Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.

2022 AMC 10, 15

Tags: geometry , cyclic quadrilateral

Quadrilateral $ABCD$ with side lengths $AB=7, BC = 24, CD = 20, DA = 15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi - b}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$? $\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$