Found problems: 85335
2024 Moldova Team Selection Test, 12
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
2022 MIG, 15
A function $f(a \tfrac bc)$ for a simplified mixed fraction $a \tfrac bc$ returns $\tfrac{a + b}{c}$. For instance, $f(2 \tfrac 57) = 1$ and $f(\tfrac45) = \tfrac45$. What is the sum of the three smallest positive rational $x$ where $f(x) = \tfrac 29$?
$\textbf{(A) }\dfrac52\qquad\textbf{(B) }\dfrac{68}{27}\qquad\textbf{(C) }\dfrac{23}{9}\qquad\textbf{(D) }\dfrac{74}{27}\qquad\textbf{(E) }\dfrac{13}4$
2004 China Team Selection Test, 3
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.
2003 China Team Selection Test, 1
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]
2016 Harvard-MIT Mathematics Tournament, 5
Let $a$, $b$, $c$, $d$, $e$, $f$ be integers selected from the set $\{1,2,\dots,100\}$, uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when $M$ is divided by $64$?
PEN A Problems, 30
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.
2008 Purple Comet Problems, 22
Let $@$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ and $c$, $(a + b)@c = (a@c) + (b@c)$ and $a@(b + c) = (a@b)@c. $ Given that $5@5=160$, find the value of $7@7$
2014 ASDAN Math Tournament, 5
Compute the smallest $9$-digit number containing all the digits $1$ to $9$ that is divisible by $99$.
2022 AMC 10, 4
In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that $1$ kilometer equals $m$ miles, and $1$ gallon equals $\ell$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon?
$\textbf{(A) } \frac{x}{100\ell m} \qquad \textbf{(B) } \frac{x\ell m}{100} \qquad \textbf{(C) } \frac{\ell m}{100x} \qquad \textbf{(D) } \frac{100}{x\ell m} \qquad \textbf{(E) } \frac{100\ell m}{x}$
2016 USAMTS Problems, 3:
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular
$n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along
the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum
possible value of $d$?
2019 Bulgaria EGMO TST, 2
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ centered at $O$, whose diagonals intersect at $H$. Let $O_1$ and $O_2$ be the circumcenters of triangles $AHD$ and $BHC$. A line through $H$ intersects $\omega$ at $M_1$ and $M_2$ and intersects the circumcircles of triangles $O_1HO$ and $O_2HO$ at $N_1$ and $N_2$, respectively, so that $N_1$ and $N_2$ lie inside $\omega$. Prove that $M_1N_1 = M_2N_2$.
PEN H Problems, 41
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.
1957 AMC 12/AHSME, 41
Given the system of equations
\[ ax \plus{} (a \minus{} 1)y \equal{} 1 \\
(a \plus{} 1)x \minus{} ay \equal{} 1.
\]
For which one of the following values of $ a$ is there no solution $ x$ and $ y$?
$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ \minus{} 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt {2}$
2019 Auckland Mathematical Olympiad, 3
Let $x$ be the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers. Prove that $x$ is odd.
2017 Online Math Open Problems, 18
Let $a,b,c$ be real nonzero numbers such that $a+b+c=12$ and \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=1.\] Compute the largest possible value of $abc-\left(a+2b-3c\right)$.
[i]Proposed by Tristan Shin[/i]
2012 Kazakhstan National Olympiad, 2
Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles
1995 Iran MO (2nd round), 3
In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.
2023 Grand Duchy of Lithuania, 3
The midpoints of the sides $BC$, $CA$ and $AB$ of triangle $ABC$ are $M$, $N$ and $P$ respectively . $G$ is the intersection point of the medians. The circumscribed circle around $BGP$ intersects the line $MP$ at the point $K$ (different than $P$).The circle circumscribed around $CGN$ intersects the line $MN$ at point $L$ (different than $N$). Prove that $\angle BAK = \angle CAL$.
1994 Turkey Team Selection Test, 1
Let $P,Q,R$ be points on the sides of $\triangle ABC$ such that $P \in [AB],Q\in[BC],R\in[CA]$ and
$\frac{|AP|}{|AB|} = \frac {|BQ|}{|BC|} =\frac{|CR|}{|CA|} =k < \frac 12$
If $G$ is the centroid of $\triangle ABC$, find the ratio $\frac{Area(\triangle PQG)}{Area(\triangle PQR)}$ .
1996 Moldova Team Selection Test, 10
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
2013 Czech-Polish-Slovak Junior Match, 6
There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.
the 16th XMO, 4
Given an integer $n$ ,For a sequence of $X$ with the number of $n$ and $Y$ with the number of $100n$ , we call it a [b]spring [/b] . We have two following rules
$\blacksquare$ Choose four adjacent character , if it is $YXXY$ , than it can be changed into $XYYX$
$\blacksquare $ Choose. four adjacent character , if it is $XYYX $ , than it can be changed into $YXXY$
If [b]spring [/b] $A$ can become $B$ using the rules , than we call they are [b][color=#3D85C6]similar [/color][/b]
Thy to find the maximum of $C$ such that there exists $C$ distinct [b]springs[/b] and they are [b][color=#3D85C6]similar[/color][/b]
2022 Putnam, A1
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y=ax+b$ intersects the curve $y=\ln(1+x^2)$ in exactly one point.
1997 Singapore Senior Math Olympiad, 1
Let $x_1,x_2,x_3,x_4, x_5,x_6$ be positive real numbers. Show that
$$\left( \frac{x_2}{x_1} \right)^5+\left( \frac{x_4}{x_2} \right)^5+\left( \frac{x_6}{x_3} \right)^5+\left( \frac{x_1}{x_4} \right)^5+\left( \frac{x_3}{x_5} \right)^5+\left( \frac{x_5}{x_6} \right)^5 \ge \frac{x_1}{x_2}+\frac{x_2}{x_4}+\frac{x_3}{x_6}+\frac{x_4}{x_1}+\frac{x_5}{x_3}+\frac{x_6}{x_5}$$
2012 National Olympiad First Round, 19
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$