Found problems: 85335
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.
2011 Purple Comet Problems, 16
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$?
2024 Mathematical Talent Reward Programme, 7
$\bigtriangleup ABC$ triangle such that $AB = AC, \angle BAC = 20 \textdegree$. $P$ is on $AB$ such that $AP = BC$, find $\frac{1}{2}\angle APC$ in degrees.
1984 AIME Problems, 15
Determine $w^2+x^2+y^2+z^2$ if
\[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]
2005 Taiwan National Olympiad, 2
Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.
2002 China Team Selection Test, 1
Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.
2016 Iran MO (3rd Round), 3
There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations?
(Observing is a one-sided relation)
2019 Serbia National Math Olympiad, 1
Find all positive integers $n, n>1$ for wich holds :
If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.
2003 Romania National Olympiad, 4
In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively.
(a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent.
(b) Knowing that $ AG_3\equal{}8,BG_1\equal{}12$ and $ CG_2\equal{}20$ compute the maximum possible value of the volume of $ ABCD$.
2005 MOP Homework, 3
For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.
BIMO 2022, 2
Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$.
What is the maximum possible value of $k$?
[i]Proposed by Ivan Chan Kai Chin[/i]