Olympiad problems

2004 Greece National Olympiad, 3

Tags: symmetry , geometry

Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.

2006 Kyiv Mathematical Festival, 1

Tags: symmetry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Squirrels $A$ and $B$ have $360$ nuts. $A$ divides these nuts into five non-empty heaps and $B$ chooses three heaps. If the total number of nuts in these heaps is divisible by the total number of nuts in other two heaps then $A$ wins. Otherwise $B$ wins. Which of the squirrels has a winning strategy?

2006 Bulgaria Team Selection Test, 1

Tags: circumcircle , parallelogram , symmetry , angle bisector , power of a point , geometry

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

2012 Math Prize For Girls Problems, 17

Tags: algebra , polynomial , symmetry

How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation \[ a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2 \, ? \]

2012 AMC 12/AHSME, 13

Tags: parabola , probability , symmetry , conic

Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $

2012 Cono Sur Olympiad, 3

Tags: symmetry , relatively prime , number theory

3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$.

2012 AMC 10, 25

Tags: probability , geometry , symmetry , integration , calculus

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

1993 All-Russian Olympiad, 2

Tags: symmetry , geometry

From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.

2012 AMC 10, 11

Tags: symmetry , rotation

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? $ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $

1999 USAMTS Problems, 4

Tags: geometry , 3d geometry , tetrahedron , symmetry , ratio , pythagorean theorem

There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.

1981 Spain Mathematical Olympiad, 3

Tags: line , 3d geometry , geometry , symmetry

Given the intersecting lines $ r$ and $s$, consider the lines $u$ and $v$ as such what: a) $u$ is symmetric to $r$ with respect to $s$, b) $v$ is symmetric to $s$ with respect to $r$ . Determine the angle that the given lines must form such that $u$ and $v$ to be coplanar.

2014 AMC 12/AHSME, 19

Tags: symmetry , algebra , polynomial , vieta

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$? $\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

2006 AMC 10, 17

Tags: probability , symmetry

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the contents of the two bags are the same? $ \textbf{(A) } \frac 1{10} \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 15 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

2007 Putnam, 1

Tags: symmetry , analytic geometry , graphing lines , slope , quadratic , geometry

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

2007 Putnam, 2

Tags: calculus , derivative , integration , symmetry , function , algebra

Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]

2007 USA Team Selection Test, 5

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection , symmetry , miquel point

Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T \equal{} BT \equal{} C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.

1997 Hungary-Israel Binational, 3

Tags: geometry , circumcircle , geometric transformation , rotation , symmetry , topology , combinatorics

Can a closed disk can be decomposed into a union of two congruent parts having no common point?

1993 Canada National Olympiad, 3

Tags: trigonometry , symmetry

In triangle $ABC,$ the medians to the sides $\overline{AB}$ and $\overline{AC}$ are perpendicular. Prove that $\cot B+\cot C\ge \frac23.$

2008 Iran Team Selection Test, 12

Tags: geometry , circumcircle , geometric transformation , reflection , symmetry , incenter , angle bisector

In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL\equal{}2R$. ($ R$ is the circumradius of $ ABC$)

1973 USAMO, 3

Tags: probability , symmetry

Three distinct vertices are chosen at random from the vertices of a given regular polygon of $ (2n\plus{}1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?

2012 Brazil Team Selection Test, 3

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]

1969 Polish MO Finals, 3

Tags: geometry , octagon , rational , symmetry

Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.

2007 Pre-Preparation Course Examination, 14

Tags: symmetry , inequalities , number theory , relatively prime

Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\] [PS: The original problem was this: Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\] But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]

1985 IMO Longlists, 97

Tags: analytic geometry , symmetry , trigonometry , function , geometry , perpendicular bisector , pythagorean theorem

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

2004 Romania Team Selection Test, 6

Tags: invariant , symmetry , quadratic , algebra , polynomial , vieta , number theory

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]