Found problems: 85335
2009 Today's Calculation Of Integral, 435
Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.
1984 National High School Mathematics League, 7
A moving point $P(x,y)$ rotate anticlockwise around unit circle, who seangular speed is $\omega$. Then how does $Q(-2xy,y^2-x^2)$ moves?
$\text{(A)}$ Rotate clockwise around unit circle, who seangular speed is $\omega$.
$\text{(B)}$ Rotate anticlockwise around unit circle, who seangular speed is $\omega$.
$\text{(C)}$ Rotate clockwise around unit circle, who seangular speed is $2\omega$.
$\text{(D)}$ Rotate anticlockwise around unit circle, who seangular speed is $2\omega$.
1999 Taiwan National Olympiad, 5
Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.
2014 Brazil National Olympiad, 1
Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
2019 BMT Spring, 13
Triangle $\vartriangle ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. $\vartriangle ABC$ has incircle $\gamma$ and circumcircle $\omega$. $\gamma$ has center at $I$. Line $AI$ is extended to hit $\omega$ at $P$. What is the area of quadrilateral $ABPC$?
2009 USAMTS Problems, 2
Let $a, b, c, d$ be four real numbers such that
\begin{align*}a + b + c + d &= 8, \\
ab + ac + ad + bc + bd + cd &= 12.\end{align*}
Find the greatest possible value of $d$.
1960 AMC 12/AHSME, 34
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
$ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 18 $
2023 Sinapore MO Open, P2
A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false?
(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.
(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.
2007 Today's Calculation Of Integral, 181
For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$
2020 Jozsef Wildt International Math Competition, W46
Let $x_1,x_2,\ldots,x_n\ge0$, $\alpha,\beta>0$, $\beta\ge\alpha$, $t\in\mathbb R$, such that $x_1^{x_2^t}\cdot x_2^{x_3^t}\cdots x_n^{x_1^t}=1$. Then prove that
$$x_1^\beta x_2^t+x_2^\beta x_3^t+\ldots+x_n^\beta x_1^t\ge x_1^\alpha x_2^t+x_2^\alpha x_3^t+\ldots+x_n^\alpha x_1^t.$$
[i]Proposed by Marius Drăgan[/i]
2013 IMC, 2
Let $\displaystyle{p,q}$ be relatively prime positive integers. Prove that
\[\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}\]
[i]Proposed by Alexander Bolbot, State University, Novosibirsk.[/i]
2007 All-Russian Olympiad, 5
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
2015 Turkey Team Selection Test, 4
Let $ABC$ be a triangle such that $|AB|=|AC|$ and let $D,E$ be points on the minor arcs $\overarc{AB}$ and $\overarc{AC}$ respectively. The lines $AD$ and $BC$ intersect at $F$ and the line $AE$ intersects the circumcircle of $\triangle FDE$ a second time at $G$. Prove that the line $AC$ is tangent to the circumcircle of $\triangle ECG$.
2018 Thailand TST, 3
Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
2022 AMC 12/AHSME, 10
What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?
$\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$
2000 Poland - Second Round, 5
Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.
2008 ITest, 95
Bored on their trip home, Joshua and Alexis decide to keep a tally of license plates they see in the other lanes: Joshua watches cars going the other way, and Alexis watches cars in the next lane.
After a few minutes, Wendy counts up the tallies and declares, "Joshua has counted $2008$ license plates, and there are $17$ license plate designs he's seen exactly $17$ times, but of Alexis's $2009$ license plates, there's none she's seen exactly $18$ times. Clearly, $17$ is the specialist number."
Michael, suspicious, pulls out the $\textit{Almanac of American License Plates}$ and notes, "According to confirmed demographic statistics, you'd only expect those numbers to be $5.4$ and $4.9$, respectively. But the $17^\text{th}$ state is weird: Joshua saw exactly $17$ of its license plates, which isn't what we'd expect."
Alexis asks, "How many Ohioan license plates did we expect to see?" and reaches for the $\textit{Almanac}$ to find out, but Michael snatches it away and says, "I'm not telling."
Alexis, disappointed, says, "I suppose that $17$ is my best guess," feeling that the answer must be pretty close to $17$.
Wendy smiles. "You can do better than that, actually. Given what Michael said and that we saw $17$ Ohioan license plates, we'd actually expect there to have been $\tfrac ab$ less than $17$."
Help Alexis. If $\tfrac ab$ is in lowest terms, find the product $ab$.
2024 Centroamerican and Caribbean Math Olympiad, 5
Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations:
\[
\begin{cases}
\sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\
\sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2
\end{cases}
\]
Find the maximum value of \(x + y\).
PEN H Problems, 18
Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.
PEN H Problems, 48
Solve the equation $x^2 +7=2^n$ in integers.
2012 Ukraine Team Selection Test, 6
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
2004 Croatia National Olympiad, Problem 1
Find all real solutions of the system of equations
$$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$
2011-2012 SDML (High School), 1
If $\left(0.67\right)^x=0.5$, then find the value of $16\cdot\left(0.67\right)^{3x}$.
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }16\qquad\text{(D) }64\qquad\text{(E) }128$
2014 IFYM, Sozopol, 6
Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?