Found problems: 85335
2006 Tournament of Towns, 1
Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)
2014 AMC 12/AHSME, 16
Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$?
$ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $
2020 Purple Comet Problems, 23
There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.
1983 AMC 12/AHSME, 24
How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal?
$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$
2005 Harvard-MIT Mathematics Tournament, 9
Compute \[ \displaystyle\sum_{k=0}^{\infty} \dfrac {4}{(4k)!}. \]
2019 Durer Math Competition Finals, 15
$ABC$ is an isosceles triangle such that $AB = AC$ and $\angle BAC = 96^o$. $D$ is the point for which $\angle ACD = 48^o$, $AD = BC$ and triangle $DAC$ is obtuse-angled. Find $\angle DAC$.
1997 Estonia National Olympiad, 5
Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]
2020 Peru Cono Sur TST., P8
Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it.
What number will be written if both play optimally?
2005 Polish MO Finals, 3
In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.
2018 Sharygin Geometry Olympiad, 10
In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?
1967 AMC 12/AHSME, 39
Given the sets of consecutive integers $\{1\}$,$ \{2, 3\}$,$ \{4,5,6\}$,$ \{7,8,9,10\}$,$\; \cdots \; $, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the $N$th set. Then $S_{21}$ equals:
$\textbf{(A)}\ 1113\qquad
\textbf{(B)}\ 4641 \qquad
\textbf{(C)}\ 5082\qquad
\textbf{(D)}\ 53361\qquad
\textbf{(E)}\ \text{none of these}$
2001 China Team Selection Test, 1
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
2001 Federal Math Competition of S&M, Problem 3
Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that
$\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$
2021 May Olympiad, 5
Bob writes $36$ consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of $36$ numbers(in the order) and writes the first $16$ results in a red paper and the last $10$ results in a blue paper. Determine if Bob can choose the $36$ integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.
2013 Stanford Mathematics Tournament, 7
A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?
2024 Caucasus Mathematical Olympiad, 6
Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold:
1) their sum is $a$;
2) the absolute value of their difference is $b$.
Find all possible values of $b$.
KoMaL A Problems 2023/2024, A. 864
Let $ABC$ be a triangle and $O$ be its circumcenter. Let $D$, $E$ and $F$ be the respective tangent points of the incircle of $\triangle ABC$, and sides $BC$, $CA$ and $AB$. Let $M$ and $N$ be the respective midpoints of sides $AB$ and $AC$. Let $M'$ and $N'$ be the respective reflections of points $M$ and $N$ across lines $DE$ and $DF$. Let lines $CM'$ and $BN'$ intersect lines $DE$ and $DF$ at points $H$ and $J$, respectively.
Prove that the points $H$, $J$ and $O$ are collinear.
[i]Proposed by Luu Dong, Vietnam[/i]
1977 Chisinau City MO, 153
Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.
2014 Serbia National Math Olympiad, 4
We call natural number $n$ [i]$crazy$[/i] iff there exist natural numbers $a$, $b >1$ such that $n=a^b+b$. Whether there exist $2014$ consecutive natural numbers among which are $2012$ [i]$crazy$[/i] numbers?
[i]Proposed by Milos Milosavljevic[/i]
1994 Korea National Olympiad, Problem 3
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$.
a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ .
b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .
1967 IMO Shortlist, 5
If $x,y,z$ are real numbers satisfying relations
\[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\]
prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.
2019 IFYM, Sozopol, 3
The perpendicular bisector of $AB$ of an acute $\Delta ABC$ intersects $BC$ and the continuation of $AC$ in points $P$ and $Q$ respectively. $M$ and $N$ are the middle points of side $AB$ and segment $PQ$ respectively. If the lines $AB$ and $CN$ intersect in point $D$, prove that $\Delta ABC$ and $\Delta DCM$ have a common orthocenter.
2018 PUMaC Geometry A, 1
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
2009 Brazil National Olympiad, 3
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
2015 India Regional MathematicaI Olympiad, 2
Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.