Found problems: 85335
2020-2021 OMMC, 9
The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$
over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$
2011 Flanders Math Olympiad, 2
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
2018 Bosnia And Herzegovina - Regional Olympiad, 2
Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct
2000 May Olympiad, 3
To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)
2016 Dutch BxMO TST, 3
Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.
(a) Prove that $\vartriangle ABC \sim \vartriangle DEF$.
(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.
2017 Harvard-MIT Mathematics Tournament, 2
Let $A$, $B$, $C$, $D$, $E$, $F$ be $6$ points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX = 3$, $BX = 2$, $CY = 4$, $DY = 10$, $EZ = 16$, and $FZ = 12$, find the perimeter of triangle $XYZ$.
1959 AMC 12/AHSME, 34
Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is:
$ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$
$\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$
2009 China Girls Math Olympiad, 3
Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n\plus{}1}\}\equal{}\{(x,y)|x,y\in \mathbb{Z}, xy\equal{}0, |x|\le n, |y|\le n\}.$
Determine the minimum of $ (P_{1}P_{2})^{2} \plus{} (P_{2}P_{3})^{2} \plus{}...\plus{} (P_{4n}P_{4n\plus{}1})^{2} \plus{} (P_{4n\plus{}1}P_{1})^{2}.$
2012 Online Math Open Problems, 20
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
[i]Author: Ray Li[/i]
1974 Putnam, A2
A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?
1967 AMC 12/AHSME, 31
Let $D=a^2+b^2+c^2$, where $a$, $b$, are consecutive integers and $c=ab$. Then $\sqrt{D}$ is:
$\textbf{(A)}\ \text{always an even integer}\qquad
\textbf{(B)}\ \text{sometimes an odd integer, sometimes not}\\
\textbf{(C)}\ \text{always an odd integer}\qquad
\textbf{(D)}\ \text{sometimes rational, sometimes not}\\
\textbf{(E)}\ \text{always irrational}$
1998 Akdeniz University MO, 1
Whichever $3$ odd numbers is given. Prove that we can find a $4.$ odd number such that, sum of squares of the these numbers is a perfect square.
1996 APMO, 1
Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.
2022 Romania National Olympiad, P1
Let $f:[0,1]\to(0,1)$ be a surjective function.
[list=a]
[*]Prove that $f$ has at least one point of discontinuity.
[*]Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity.
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
2016 Indonesia TST, 4
In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.
2015 Grand Duchy of Lithuania, 4
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions:
$\bullet$ gcd $(a, b, c)=1$,
$\bullet$ gcd $(a, b + c)>1$,
$\bullet$ gcd $(b, c + a)>1$,
$\bullet$ gcd $(c, a + b)>1$.
a) Is it possible that $a + b + c = 2015$?
b) Determine the minimum possible value that the sum $a+ b+ c$ can take.
2003 Flanders Math Olympiad, 4
Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points)
Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.
1993 All-Russian Olympiad Regional Round, 9.3
Points $M$ and $N$ are chosen on the sides $AB$ and BC of a triangle $ABC$. The segments $AN$ and $CM$ meet at $O$ such that $AO =CO$. Is the triangle $ABC$ necessarily isosceles, if
(a) $AM = CN$?
(b) $BM = BN$?
2022 Bulgarian Spring Math Competition, Problem 8.4
Let $p = (a_{1}, a_{2}, \ldots , a_{12})$ be a permutation of $1, 2, \ldots, 12$.
We will denote \[S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|\]We'll call $p$ $\textit{optimistic}$ if $a_{i} > \min(a_{i-1}, a_{i+1})$ $\forall i = 2, \ldots, 11$.
$a)$ What is the maximum possible value of $S_{p}$. How many permutations $p$ achieve this maximum?$\newline$
$b)$ What is the number of $\textit{optimistic}$ permtations $p$?
$c)$ What is the maximum possible value of $S_{p}$ for an $\textit{optimistic}$ $p$? How many $\textit{optimistic}$ permutations $p$ achieve this maximum?
2011 F = Ma, 5
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit?
(A) $\text{exactly 0 m/s}^2$
(B) $\text{0.006 m/s}^2$
(C) $\text{0.6 m/s}^2$
(D) $\text{6 m/s}^2$
(E) $\text{10 m/s}^2$
2011 Middle European Mathematical Olympiad, 2
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
2018 Online Math Open Problems, 24
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
[list]
[*] $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$, and
[*] $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.
[/list]
Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
[i]Proposed by Ankan Bhattacharya[/i]
2021 LMT Fall, 10
Convex cyclic quadrilateral $ABCD$ satisfies $AC \perp BD$ and $AC$ intersects $BD$ at $H$. Let the line through $H$ perpendicular to $AD$ and the line through $H$ perpendicular to $AB$ intersect $CB$ and $CD$ at $P$ and $Q$, respectively. The circumcircle of $\triangle CPQ$ intersects line $AC$ again at $X \ne C$. Given that $AB=13$, $BD=14$, and $AD=15$, the length of $AX$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a,b,c,x,y,$ and $z$ be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If $xy+yz+xz=67$ and $x+y+z=2010$, find the value of $xyz$.
2006 QEDMO 2nd, 13
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any two reals $x$ and $y$, we have
$f\left( f\left( x+y\right) \right) +xy=f\left( x+y\right) +f\left(
x\right) f\left( y\right) $.