Found problems: 85335
2010 Today's Calculation Of Integral, 557
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
2017 Iberoamerican, 1
For every positive integer $n$ let $S(n)$ be the sum of its digits. We say $n$ has a property $P$ if all terms in the infinite secuence $n, S(n), S(S(n)),...$ are even numbers, and we say $n$ has a property $I$ if all terms in this secuence are odd. Show that for, $1 \le n \le 2017$ there are more $n$ that have property $I$ than those who have $P$.
2012 Polish MO Finals, 5
Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.
2015 Moldova Team Selection Test, 2
Consider a triangle $\triangle ABC$, let the incircle centered at $I$ touch the sides $BC,CA,AB$ at points $D,E,F$ respectively. Let the angle bisector of $\angle BIC$ meet $BC$ at $M$, and the angle bisector of $\angle EDF$ meet $EF$ at $N$. Prove that $A,M,N$ are collinear.
2015 Harvard-MIT Mathematics Tournament, 10
Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of [b]negative slope[/b] tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a [b]unique[/b] point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.
Putnam 1938, A6
A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?
2017 OMMock - Mexico National Olympiad Mock Exam, 5
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
1998 Baltic Way, 17
Let $n$ and $k$ be positive integers. There are $nk$ objects (of the same size) and $k$ boxes, each of which can hold $n$ objects. Each object is coloured in one of $k$ different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.
2005 Brazil Undergrad MO, 4
Let $a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}$ and $a_1=1$. Show that $\sum^{\infty}_{n=1}{\frac{1}{n a_n}}$ converge.
1992 IMO Longlists, 13
Let $ABCD$ be a convex quadrilateral such that $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$
1949-56 Chisinau City MO, 13
Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$
2025 Spain Mathematical Olympiad, 2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
2021 JHMT HS, 2
Compute the smallest positive integer $n$ such that $\int_{0}^{n} \lfloor x\rfloor\,dx$ is at least $2021.$
2020 JBMO Shortlist, 7
Prove that there doesn’t exist any prime $p$ such that every power of $p$ is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).
1999 Junior Balkan Team Selection Tests - Romania, 4
Let be a convex quadrilateral $ ABCD. $ On the semi-straight line extension of $ AB $ in the direction of $ B, $ put $ A_1 $ such that $ AB=BA_1. $ Similarly, define $ B_1,C_1,D_1, $ for the other three sides.
[b]a)[/b] If $ E,E_1,F,F_1 $ are the midpoints of $ BC,A_1B_1,AD $ respectively, $ C_1,D_1, $ show that $ EE_1=FF_1. $
[b]b)[/b] Delete everything, but $ A_1,B_1,C_1,D_1. $ Now, find a way to construct the initial quadrilateral.
[i]Vasile Pop[/i]
2016 Rioplatense Mathematical Olympiad, Level 3, 1
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.
2013 Balkan MO Shortlist, G3
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$. A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$, respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively. Prove that $ABCD$ is a parallelogram.
2008 China Western Mathematical Olympiad, 3
Given an integer $ m\geq$ 2, m positive integers $ a_1,a_2,...a_m$. Prove that there exist infinitely many positive integers n, such that $ a_{1}1^{n} \plus{} a_{2}2^{n} \plus{} ... \plus{} a_{m}m^{n}$ is composite.
2006 AMC 8, 23
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 5$
1995 IberoAmerican, 1
In a $m\times{n}$ grid are there are token. Every token [i]dominates [/i] every square on its same row ($\leftrightarrow$), its same column ($\updownarrow$), and diagonal ($\searrow\hspace{-4.45mm}\nwarrow$)(Note that the token does not \emph{dominate} the diagonal ($\nearrow\hspace{-4.45mm}\swarrow$), determine the lowest number of tokens that must be on the board to [i]dominate [/i] all the squares on the board.
2012 Flanders Math Olympiad, 1
Our class decides to have a alpha - beta - gamma tournament. This party game is always played in groups of three. Any possible combination of three players (three students or two students and the teacher) plays the game $1$ time. The player who wins gets $1$ point. The two losers get no points. At the end of the tournament, miraculously, all students have as many points. The teacher has $3$ points. How many students are there in our class?
2022 JBMO Shortlist, G3
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
2010 Stanford Mathematics Tournament, 4
If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$
2021 MIG, 12
Jo claims that any two triangles, both having a perimeter of four, are congruent. Jann claims that two circles, both having a circumference of $4\pi$, are congruent. Julia claims that two squares, both having a perimeter of four, are congruent. Which of these students are correct?
$\textbf{(A) }\text{Jo}\qquad\textbf{(B) }\text{Jann}\qquad\textbf{(C) }\text{Julia}\qquad\textbf{(D) }\text{Jo, Julia}\qquad\textbf{(E) }\text{Jann, Julia}$
2007 China Team Selection Test, 2
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.