Olympiad problems

2010 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: trigonometry

Prove that there is no real number $x$ satisfying both equations \begin{align*}2^x+1=2\sin x \\ 2^x-1=2\cos x.\end{align*}

XMO (China) 2-15 - geometry, 7.1

Tags: geometry , perpendicular

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

2022 Olympic Revenge, Problem 3

Tags: combinatorics , chessboard

positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least $Cn$ cells have the same color. a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$. b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.

1974 AMC 12/AHSME, 15

Tags:

If $ x<\minus{}2$ then $ |1\minus{}|1\plus{}x|$ $ |$ equals $ \textbf{(A)}\ 2\plus{}x \qquad \textbf{(B)}\ \minus{}2\minus{}x \qquad \textbf{(C)}\ x \qquad \textbf{(D)}\ \minus{}x \qquad \textbf{(E)}\ \minus{}2$

1994 Vietnam Team Selection Test, 2

Tags: function , algebra , logarithm

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

2013 BMT Spring, 10

Tags: combinatorics

In a far away kingdom, there exist $k^2$ cities subdivided into k distinct districts, such that in the $i^ {th}$ district, there exist $2i - 1$ cities. Each city is connected to every city in its district but no cities outside of its district. In order to improve transportation, the king wants to add $k - 1$ roads such that all cities will become connected, but his advisors tell him there are many ways to do this. Two plans are different if one road is in one plan that is not in the other. Find the total number of possible plans in terms of $k$.

2023 ELMO Shortlist, G8

Tags: geometry

Convex quadrilaterals $ABCD$, $A_1B_1C_1D_1$, and $A_2B_2C_2D_2$ are similar with vertices in order. Points $A$, $A_1$, $B_2$, $B$ are collinear in order, points $B$, $B_1$, $C_2$, $C$ are collinear in order, points $C$, $C_1$, $D_2$, $D$ are collinear in order, and points $D$, $D_1$, $A_2$, $A$ are collinear in order. Diagonals $AC$ and $BD$ intersect at $P$, diagonals $A_1C_1$ and $B_1D_1$ intersect at $P_1$, and diagonals $A_2C_2$ and $B_2D_2$ intersect at $P_2$. Prove that points $P$, $P_1$, and $P_2$ are collinear. [i]Proposed by Holden Mui[/i]