Found problems: 85335
2011 Today's Calculation Of Integral, 708
Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.
2016 South African National Olympiad, 4
For which integers $n \geq 2$ is it possible to draw $n$ straight lines in the plane in such a way that there are at least $n - 2$ points where exactly three of the lines meet?
2021 Princeton University Math Competition, B2
Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$
2007 Today's Calculation Of Integral, 237
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
2018 Hanoi Open Mathematics Competitions, 4
Let $a = (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3)$
$b = (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3)$
The difference $a - b$ belongs to the set:
A. $(-\infty,-4)$ B. $[-4,0)$ C.$\{0\}$ D. $(0,4]$ E. $(4,\infty)$
1997 National High School Mathematics League, 11
$ABCDEF$ is a regular hexagon. A frog sarts jumping at $A$, each time it can jump to one of the two adjacent points. If the frog jump to $D$ in no more than five times, it stops. After five jumpings, if the frog hasn't jumped to $D$ yet, it will stop as well. Then the number of different ways to jump is________.
2001 Singapore MO Open, 4
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.
(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
1997 Israel National Olympiad, 1
Find all real solutions to the system of equations
$$\begin{cases} x^2 +y^2 = 6z \\
y^2 +z^2 = 6x \\
z^2 +x^2 = 6y \end{cases}$$
2009 Indonesia TST, 2
Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.
2014 Switzerland - Final Round, 1
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.
1950 Moscow Mathematical Olympiad, 176
Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$
1985 AMC 8, 3
$ \frac{10^7}{5 \times 10^4}\equal{}$
\[ \textbf{(A)}\ .002 \qquad
\textbf{(B)}\ .2 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 200 \qquad
\textbf{(E)}\ 2000
\]
2012 Today's Calculation Of Integral, 819
For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$
Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$
2019 BMT Spring, 3
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $. Find the sum of the sum of the coordinates of all such points.
2025 AIME, 6
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$
2012 Math Prize For Girls Problems, 16
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
2019 Costa Rica - Final Round, 3
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$
2013 ELMO Problems, 3
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2012 IMO Shortlist, A6
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded.
[i]Proposed by Palmer Mebane, United States[/i]
2015 Taiwan TST Round 2, 2
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
1997 Iran MO (3rd Round), 2
In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$.
1985 Polish MO Finals, 4
$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$
2023 Princeton University Math Competition, 15
15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ?
2024 All-Russian Olympiad Regional Round, 11.5
The equation $$t^4+at^3+bt^2=(a+b)(2t-1)$$ has $4$ positive real roots $t_1<t_2<t_3<t_4$. Show that $t_1t_4>t_2t_3$.
MOAA Gunga Bowls, 2021.12
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
[i]Proposed by Andrew Wen[/i]