Olympiad problems

2020 Simon Marais Mathematics Competition, B4

[i]The following problem is open in the sense that no solution is currently known to part (b).[/i] Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices. We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct. (a) show that if $n-1$ is prime then $n$ is taut. (b) Which integers $n\geq 2$ are taut?

1980 IMO, 5

Tags: combinatorics , geometry , function , 3d geometry , sphere

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

2004 Oral Moscow Geometry Olympiad, 2

Tags: broken line , geometry

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

2004 District Olympiad, 4

Tags: trapezoid , geometry , trigonometry , 3d geometry , palnes , right angle , angle

In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$. At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine: a) the cosine of the angle between the planes $MNP$ and $ABC$. b) the distance from $D$ to the plane $MNP$

1994 Hungary-Israel Binational, 3

Tags: geometry

Three given circles have the same radius and pass through a common point $ P$. Their other points of pairwise intersections are $ A$, $ B$, $ C$. We define triangle $ A'B'C'$, each of whose sides is tangent to two of the three circles. The three circles are contained in $ \triangle A'B'C'$. Prove that the area of $ \triangle A'B'C'$ is at least nine times the area of $ \triangle ABC$

2005 AMC 10, 11

Tags: geometry , 3d geometry

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

2017 BMT Spring, 5

Tags: geometry

Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?

2019 Belarus Team Selection Test, 8.1

Tags: symmetry , angle chasing , miquel point , geometry

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

2016 Online Math Open Problems, 9

Tags: geometry

In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$? [i]Proposed by James Lin[/i]

1998 National Olympiad First Round, 12

Tags: trigonometry , geometry , ratio , perimeter

In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval? $\textbf{(A)}\ \frac{2\sqrt{2} \plus{}1}{4} \qquad\textbf{(B)}\ \frac{\sqrt{2} \plus{}1}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad\textbf{(D)}\ \sqrt{2} \minus{}1 \qquad\textbf{(E)}\ \frac{\sqrt{2} \minus{}1}{2}$

1961 All Russian Mathematical Olympiad, 006

Tags: geometry , equilateral , locus

a) Points $A$ and $B$ move uniformly and with equal angle speed along the circumferences with $O_a$ and $O_b$ centres (both clockwise). Prove that a vertex $C$ of the equilateral triangle $ABC$ also moves along a certain circumference uniformly. b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.

2014 Contests, 3

Tags: geometry , parallelogram

A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

2016 Oral Moscow Geometry Olympiad, 1

Tags: geometry , incenter

The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the altitudes $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the midpoint of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.

1960 IMO Shortlist, 3

Tags: circumcircle , geometry , trigonometry , trigonometric identities

In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]

1976 Bulgaria National Olympiad, Problem 3

Tags: tetrahedron , 3d geometry , geometry

In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron. [i]J. Tabov[/i]

2004 Bundeswettbewerb Mathematik, 2

Tags: reflection , geometry , geometric transformation , parameterization , integration , function , perimeter

Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords. Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.

2008 China Northern MO, 1A

Tags: geometry , trapezoid , tangential , equal angles

As shown in figure , $\odot O$ is the inscribed circle of trapezoid $ABCD$, and the tangent points are $E, F, G, H$, $AB \parallel CD$. The line passing through$ B$, parallel to $AD$ intersects extension of $DC$ at point $P$. The extension of $AO$ intersects $CP$ at point $Q$. If $AE=BE$ , prove that $\angle CBQ = \angle PBQ$. [img]https://cdn.artofproblemsolving.com/attachments/d/2/7c3a04bb1c59bc6d448204fd78f553ea53cb9e.png[/img]

2015 Korea National Olympiad, 2

Tags: geometry , circumcircle

Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.

2012 Balkan MO, 1

Tags: reflection , circumcircle , geometry

Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$. Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.

2018 ITAMO, 2

Tags: geometry

$2.$Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let Ω be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between Ω and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$. Prove that $GS$ is perpendicular to $BC$

2018 Costa Rica - Final Round, G5

Tags: geometry , semicircle , tangent circle

In the accompanying figure, semicircles with centers$ A$ and $B$ have radii $4$ and $2$, respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$. Also the semicircles with centers $ A$ and $ B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/281b335f6a2d6230a5b79060e6d85d6ca6f06c.png[/img]

2004 Harvard-MIT Mathematics Tournament, 8

Tags: geometry

A triangle has side lengths $18$, $24$, and $30$. Find the area of the triangle whose vertices are the incenter, circumcenter, and centroid of the original triangle.

2013 BMT Spring, 5

Tags: geometry

Points $A$ and $B$ are fixed points in the plane such that $AB = 1$. Find the area of the region consisting of all points $P$ such that $\angle APB > 120^o$

2023/2024 Tournament of Towns, 4

Tags: geometry

4. Given is an acute-angled triangle $A B C, H$ is its orthocenter. Let $P$ be an arbitrary point inside (and not on the sides) of the triangle $A B C$ that belongs to the circumcircle of the triangle $A B H$. Let $A^{\prime}, B^{\prime}$, $C^{\prime}$ be projections of point $P$ to the lines $B C, C A, A B$. Prove that the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime}$ passes through the midpoint of segment $C P$. Alexey Zaslavsky

2011 Czech and Slovak Olympiad III A, 5

Tags: circumcircle , geometry

In acute triangle ABC, which is not equilateral, let $P$ denote the foot of the altitude from $C$ to side $AB$; let $H$ denote the orthocenter; let $O$ denote the circumcenter; let $D$ denote the intersection of line $CO$ with $AB$; and let $E$ denote the midpoint of $CD$. Prove that line $EP$ passes through the midpoint of $OH$.