Olympiad problems

2013 Miklós Schweitzer, 6

Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation \[ \displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+\] Prove that if for all ${x,y \in \mathcal A_+}$ we have \[ \displaystyle (x\circ y)\circ y = x \circ (y \circ y), \] then ${\mathcal A}$ is commutative. [i]Proposed by Lajos Molnár[/i]

1968 AMC 12/AHSME, 11

Tags: geometry , ratio

If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is: $\textbf{(A)}\ 16:9 \qquad \textbf{(B)}\ 9:16 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:4 \qquad \textbf{(E)}\ \text{None of these} $

1993 Korea - Final Round, 1

Tags: rectangle , combinatorics , geometry

Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).

2017 USA TSTST, 5

Tags: geometry , monge d alembert

Let $ABC$ be a triangle with incenter $I$. Let $D$ be a point on side $BC$ and let $\omega_B$ and $\omega_C$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC$ at points $E$ and $F$, respectively. Let $P$ be the intersection of segment $AD$ with the line joining the centers of $\omega_B$ and $\omega_C$. Let $X$ be the intersection point of lines $BI$ and $CP$ and let $Y$ be the intersection point of lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ meet on the incircle of $\triangle ABC$. [i]Proposed by Ray Li[/i]

2011 NZMOC Camp Selection Problems, 2

Tags: diameter , geometry

Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

1991 Vietnam Team Selection Test, 1

Tags: geometry , parallelogram , circumcircle

1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions: [b]I.[/b] The distance between any two points $\in S$ is not greater than 1. [b]II.[/b] For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$. [b]III. [/b] For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$. Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.

1969 Poland - Second Round, 5

Tags: geometry , projection

Prove that if, in parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of every figure is the figure congruent to it.

2013 BMT Spring, 19

Tags: geometry

Equilateral triangle $ABC$ is inscribed in a circle. Chord $AD$ meets $BC$ at $E$. If $DE = 2013$, how many scenarios exist such that both $DB$ and $DC$ are integers (two scenarios are different if $AB$ is different or $AD$ is different)?

1962 Bulgaria National Olympiad, Problem 4

Tags: triangle , geometry

There are given a triangle and some internal point $P$. $x,y,z$ are distances from $P$ to the vertices $A,B$ and $C$. $p,q,r$ are distances from $P$ to the sides $BC,CA,AB$ respectively. Prove that: $$xyz\ge(q+r)(r+p)(p+q).$$

2000 Balkan MO, 2

Tags: geometry

Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$. Let $E$ be a point on segment $AD$ and $M$ its projection on $BC$. If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.

2001 National Olympiad First Round, 22

Tags: rectangle , geometry

A ladder is formed by removing some consecutive unit squares of a $10\times 10$ chessboard such that for each $k-$th row ($k\in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have? $ \textbf{(A)}\ 625 \qquad\textbf{(B)}\ 715 \qquad\textbf{(C)}\ 1024 \qquad\textbf{(D)}\ 1512 \qquad\textbf{(E)}\ \text{None of the preceding} $

1977 Poland - Second Round, 2

Tags: geometry , inequalities , geometric inequality

Let $X$ be the interior point of triangle $ABC$. prove that the product of the distances of point $ X $ from the vertices $ A, B, C $ is at least eight times greater than the product of the distances of this point from the lines $ AB, BC, CA $.

1999 Moldova Team Selection Test, 11

Tags: geometry

Let $ABC$ be a triangle. Show that there exists a lin $l$ in the plane of $ABC$ such that the overlapping area of $ABC$ and $A^{'}B^{'}C^{'}$, the symmetric of $ABC$ with respect to $l$, is greater than $\frac{2}{3}$ of area of $ABC$.

1996 Cono Sur Olympiad, 1

Tags: rectangle , square , area , geometry

In the following figure, the largest square is divided into two squares and three rectangles, as shown: The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]

2021 Durer Math Competition Finals, 16

Tags: geometry , angle , arithmetic sequence

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length $1$, then its perimeter is $a + \sqrt{b}4$, where $ a$ and $b$ are positive integers. What is the value of $a + b$?

2013 Oral Moscow Geometry Olympiad, 6

Tags: trapezoid , incircle , arc midpoint , concurrency , geometry

The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.

Durer Math Competition CD 1st Round - geometry, 2017.C1

Tags: angle , decagon , geometry

The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?

2006 National Olympiad First Round, 21

Tags: incenter , geometry

Let $ABC$ be a triangle with $m(\widehat A) = 70^\circ$ and the incenter $I$. If $|BC|=|AC|+|AI|$, then what is $m(\widehat B)$? $ \textbf{(A)}\ 35^\circ \qquad\textbf{(B)}\ 36^\circ \qquad\textbf{(C)}\ 42^\circ \qquad\textbf{(D)}\ 45^\circ \qquad\textbf{(E)}\ \text{None of above} $

2007 Hanoi Open Mathematics Competitions, 13

Tags: geometry

Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.

1997 Korea National Olympiad, 7

Tags: geometry

Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.

2010 Tournament Of Towns, 1

Tags: geometry

A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they dene a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.

2019 HMNT, 7

Tags: geometry

Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible perimeters of the pentagon he obtains is exactly the interval $[a, b)$, where $a$ and $b$ are positive real numbers. Compute $\frac{a}{b}$ .

2001 Kazakhstan National Olympiad, 2

Tags: centroid , orthocenter , incenter , geometry

In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.

2009 Sharygin Geometry Olympiad, 6

Tags: geometry

Find the locus of excenters of right triangles with given hypotenuse.

2024 Thailand TST, 1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.