Olympiad problems

1995 All-Russian Olympiad, 3

Tags: quadratic , polynomial , symmetry , function , complex number , algebra

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

2008 Harvard-MIT Mathematics Tournament, 2

Tags: symmetry

Let $ S \equal{} \{1,2,\ldots,2008\}$. For any nonempty subset $ A\in S$, define $ m(A)$ to be the median of $ A$ (when $ A$ has an even number of elements, $ m(A)$ is the average of the middle two elements). Determine the average of $ m(A)$, when $ A$ is taken over all nonempty subsets of $ S$.

2009 All-Russian Olympiad, 2

Tags: geometry , circumcircle , incenter , ratio , symmetry , geometric transformation

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

2014 India Regional Mathematical Olympiad, 4

Tags: number theory , least common multiple , inequalities , symmetry , vieta , system of equations , algebra

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

2012 Singapore Senior Math Olympiad, 1

Tags: incenter , symmetry , geometry

A circle $\omega$ through the incentre$ I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of$ BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.

2005 Cono Sur Olympiad, 3

Tags: symmetry , analytic geometry , combinatorics

On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.

2000 Italy TST, 2

Tags: linear algebra , matrix , symmetry , angle bisector , geometry

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

2015 Sharygin Geometry Olympiad, P14

Tags: geometry , circles , symmetry

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$. Segment $A_1A_2$ is a diameter of a circle $\alpha$. Circles $\beta$ and $\gamma$ are defined similarly. Prove that these three circles have two common points.

1970 AMC 12/AHSME, 33

Tags: symmetry

Find the sum of the digits of all numerals in the sequence $1,2,3,4,\cdots ,10000$. $\textbf{(A) }180,001\qquad\textbf{(B) }154,756\qquad\textbf{(C) }45,001\qquad\textbf{(D) }154,755\qquad \textbf{(E) }270,001$

2009 Today's Calculation Of Integral, 401

Tags: calculus , integration , trigonometry , symmetry , limit , calculus computations

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

2008 Putnam, B3

Tags: trigonometry , symmetry , inequalities , 3d geometry , sphere , geometry

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

2011 IMO Shortlist, 4

Tags: geometry , circumcircle , symmetry , homothety , radical axis , marvio

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

1980 IMO Longlists, 21

Tags: symmetry , geometry

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

2010 Contests, 1

Tags: symmetry , subset , bounding , combinatorics

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

2009 IMO Shortlist, 3

Tags: symmetry , triangle , parallelogram , geometry

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

2009 Balkan MO Shortlist, C1

Tags: symmetry , geometry , rectangle , combinatorics

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

2006 India IMO Training Camp, 1

Tags: greatest common divisor , modular arithmetic , symmetry , number theory

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

2010 AMC 12/AHSME, 18

Tags: analytic geometry , symmetry , geometry , geometric transformation , reflection

A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$

2006 Hong kong National Olympiad, 3

Tags: circumcircle , symmetry , power of a point , radical axis , geometry

A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.

2014 AIME Problems, 14

Tags: quadratic , algebra , polynomial , symmetry , sfft , linear algebra , contrived

Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.

1975 Chisinau City MO, 110

Tags: octagon , symmetry , geometry

Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.

2002 Putnam, 4

Tags: linear algebra , matrix , symmetry , putnam games

In Determinant Tic-Tac-Toe, Player $1$ enters a $1$ in an empty $3 \times 3$ matrix. Player $0$ counters with a $0$ in a vacant position and play continues in turn intil the $ 3 \times 3 $ matrix is completed with five $1$’s and four $0$’s. Player $0$ wins if the determinant is $0$ and player $1$ wins otherwise. Assuming both players pursue optimal strategies, who will win and how?

2004 USAMO, 1

Tags: inequalities , geometry , incenter , inradius , trigonometry , symmetry , trig identities

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

2012 Turkey Junior National Olympiad, 2

Tags: geometry , circumcircle , ratio , trapezoid , symmetry , geometric transformation , reflection

In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.

1994 AIME Problems, 6

Tags: analytic geometry , geometry , rhombus , geometric transformation , reflection , quadratic , symmetry

The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed?