Olympiad problems

2009 Italy TST, 2

Tags: geometry , circumcircle , trapezoid , geometric transformation , reflection , power of a point , radical axis

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

1991 Vietnam National Olympiad, 2

Tags: geometry , circumcircle , inequalities

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that: $\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

1935 Moscow Mathematical Olympiad, 012

Tags: geometry , 3d geometry , cone , pyramid , angle

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

1988 Bulgaria National Olympiad, Problem 6

Tags: polynomial , fe , functional equation , algebra

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

Denmark (Mohr) - geometry, 2001.3

Tags: geometry , min , square

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

2022 HMNT, 7

Tags:

All positive integers whose binary representations (excluding leading zeroes) have at least as many $1$’s as $0$’s are put in increasing order. Compute the number of digits in the binary representation of the $200$th number.

2010 Argentina National Olympiad, 1

Tags: combinatorics

Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?

2004 Federal Math Competition of S&M, 2

Tags: geometry

Let $r$ be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed $3r$

2022 BMT, Tie 1

Tags: algebra

For all $a$ and $b$, let $a\clubsuit b = 3a + 2b + 1$. Compute $c$ such that $(2c)\clubsuit (5\clubsuit (c + 3)) = 60$.

2014 AMC 8, 11

Tags: asymptote , analytic geometry , probability

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$