Found problems: 85335
2022 Purple Comet Problems, 30
There is a positive integer s such that there are s solutions to the equation
$64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$
where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$.
1954 AMC 12/AHSME, 30
$ A$ and $ B$ together can do a job in $ 2$ days; $ B$ and $ C$ can do it in four days; and $ A$ and $ C$ in $ 2\frac{2}{5}$ days. The number of days required for $ A$ to do the job alone is:
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 2.8$
2018 Denmark MO - Mohr Contest, 4
A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the$ 99$ fractions$$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$?
2022 Belarusian National Olympiad, 10.6
Circles $\omega_1$ and $\omega_2$ intersect at $X$ and $Y$. Through point $Y$ two lines pass, one of which intersects $\omega_1$ and $\omega_2$ for the second time at $A$ and $B$, and the other at $C$ and $D$. Line $AD$ intersects for the second time circles $\omega_1$ and $\omega_2$ at $P$ and $Q$. It turned out that $YP=YQ$
Prove that the circumcircles of triangles $BCY$ and $PQY$ are tangent to each other.
2019 CIIM, Problem 3
Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers.
For $m \geq 1$, we define:
\[ X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}. \]
Show that
\[\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.\]
2018 JBMO Shortlist, A1
Let $x,y,z$ be positive real numbers . Prove:
$\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}$
2013 NIMO Problems, 8
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Michael Ren[/i]
2012 India IMO Training Camp, 2
Let $a\ge b$ and $c\ge d$ be real numbers. Prove that the equation
\[(x+a)(x+d)+(x+b)(x+c)=0\]
has real roots.
2012 AMC 10, 14
Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
$ \textbf{(A)}\ 480
\qquad\textbf{(B)}\ 481
\qquad\textbf{(C)}\ 482
\qquad\textbf{(D)}\ 483
\qquad\textbf{(E)}\ 484
$
2001 National High School Mathematics League, 10
The solution to inequality $\left|\frac{1}{\log_{\frac{1}{2}}x}+2\right|>\frac{3}{2}$ is________(express answer with a set).
2003 National Olympiad First Round, 27
A finite number of circles are placed into a $1 \times 1$ square. Let $C$ be the sum of the perimeters of the circles. For how many $C$s from $C=\dfrac {43}5$, $9$, $\dfrac{91}{10}$, $\dfrac{19}{2}$, $10$, we can definitely say there exists a line cutting four of the circles?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
2024 Malaysia IMONST 2, 2
Jia Herng has a circle $\omega$ with center $O$, and $P$ is a point outside of $\omega$. Let $PX$ and $PY$ are two lines tangent to $\omega$ at $X$ and $Y$ , and $Q$ is a point on segment $PX$. Let $R$ is a point on the ray $PY$ beyond $Y$ such that $QX = RY$.
Help Jia Herng prove that the points $O$, $P$, $Q$, $R$ are concyclic.
2000 National Olympiad First Round, 4
What is the sum of real roots of $(x\sqrt{x})^x = x^{x\sqrt{x}}$?
$ \textbf{(A)}\ \frac{18}{7}
\qquad\textbf{(B)}\ \frac{71}{4}
\qquad\textbf{(C)}\ \frac{9}{4}
\qquad\textbf{(D)}\ \frac{24}{19}
\qquad\textbf{(E)}\ \frac{13}{4}
$
2007 Princeton University Math Competition, 10
if $x$, $y$, and $z$ are real numbers such that $ x^2 + z^2 = 1 $ and $ y^2 + 2y \left( x + z \right) = 6 $, find the maximum value of $ y \left( z - x \right) $.
2021 Iran Team Selection Test, 1
In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$.
Proposed by [i]Alireza Haghi[/i]
2014 District Olympiad, 4
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}$ be a strictly increasing function. Prove that:
[list=a]
[*]There exists a decreasing sequence of positive real numbers, $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $y_{n}\leq2y_{f(n)}$, for all $n\in\mathbb{N}$.
[*]If $(x_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $x_{n}\leq y_{n} \leq2y_{f(n)}$, for all $n\in\mathbb{N}$.[/list]
1999 Slovenia National Olympiad, Problem 2
The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?
2018 Tuymaada Olympiad, 3
A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$.
[i]Proposed by A. Antropov[/i]
2018 JHMT, 6
$\vartriangle ABC$ is inscribed in a unit circle. The three angle bisectors of $A$,$B$,$C$ are extended to intersect the circle at $A_1$,$B_1$,$C_1$, respectively. Find $$\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}.$$
2018 Brazil Undergrad MO, 24
What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$
2016 China Team Selection Test, 1
$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.
2024 Belarusian National Olympiad, 8.2
Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$
For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$
where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$.
Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd
[i]M. Zorka[/i]
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
1952 Poland - Second Round, 4
Prove that if the numbers $ a $, $ b $, $ c $ satisfy the equation
$$
\frac{1}{ab} + \frac{1}{bc} +\frac{1}{ca} = \frac{1}{ab + bc + ca},$$
then two of them are opposite numbers.
1999 Denmark MO - Mohr Contest, 4
Nanna and Sofie move in the same direction along two parallel paths, which are $200$ meters apart. Nanna's speed is $3$ meters per second, Sofie's only $1$ meter per second. A tall, cylindrical building with a diameter of $100$ meters is located in the middle between the two paths. Since the building first once the line of sight breaks between the girls, their distance between them is $200$ metres. How long will it be before the two girls see each other again?