Found problems: 85335
Durer Math Competition CD Finals - geometry, 2009.D3
What is the area of the letter $O$ made by Dürer? The two circles have a unit radius. Their centers, or the angle of a triangle formed by an intersection point of the circles is $30^o$.
[img]https://cdn.artofproblemsolving.com/attachments/b/c/fe052393871a600fc262bd60047433972ae1be.png[/img]
2007 All-Russian Olympiad, 1
Prove that for $k>10$ Nazar may replace in the following product some one $\cos$ by $\sin$ so that the new function $f_{1}(x)$ would satisfy inequality $|f_{1}(x)|\le 3\cdot 2^{-1-k}$ for all real $x$.
\[f(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x \]
[i]N. Agakhanov[/i]
2011 Peru IMO TST, 6
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
2015 Canadian Mathematical Olympiad Qualification, 6
Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median.
Prove that $3AD + AF > 4AE$.
2011 Indonesia TST, 2
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
1983 IMO Longlists, 7
Find all numbers $x \in \mathbb Z$ for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2022-23 IOQM India, 9
Two sides of an integer sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer $y$ such that $18,x,y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have.
2013 Costa Rica - Final Round, G3
Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.
2019 Harvard-MIT Mathematics Tournament, 4
Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.
2024 Lusophon Mathematical Olympiad, 6
A positive integer $n$ is called $oeirense$ if there exist two positive integers $a$ and $b$, not necessarily distinct, such that $n=a^2+b^2$.
Determine the greatest integer $k$ such that there exist infinitely many positive integers $n$ such that $n$, $n+1$, $\dots$, $n+k$ are oeirenses.
2004 Thailand Mathematical Olympiad, 16
What are last three digits of $2^{2^{2004}}$ ?
2014 German National Olympiad, 6
Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
1998 Gauss, 20
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one
red edge. What is the smallest number of red edges?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
MOAA Team Rounds, 2022.2
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
2007 Chile National Olympiad, 3
Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
1993 All-Russian Olympiad, 1
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
2002 JBMO ShortLists, 11
Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$.
III Soros Olympiad 1996 - 97 (Russia), 9.1
Without using a calculator, find out which number is greater:
$$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$
1999 Harvard-MIT Mathematics Tournament, 6
You want to sort the numbers 5 4 3 2 1 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, [i]6 5 3[/i] 4 2 1 -> 4 2 [i]6 5 3[/i] 1 is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 1 2 3 4 5?
1968 All Soviet Union Mathematical Olympiad, 106
Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.
2020 Balkan MO Shortlist, A2
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]