Found problems: 85335
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 $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$