Found problems: 85335
Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$
The figure below is used to fold into a pyramid, and consists of four equilateral triangles erected around a square with area nine. What is the length of the dashed path shown?
[asy]
real r = 1/2 * 3^(1/2);
size(45);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0,0)--(-r,0.5)--(0,1)--(0.5,1+r)--(1,1)--(1+r,0.5)--(1,0)--(0.5,-r)--cycle,dashed);
[/asy]
$\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }24\qquad\textbf{(E) }27$
Find the sum of all integers $x$ for which there is an integer $y$, such that $x^3-y^3=xy+61$.
$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.
There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/i] to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.
Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.
A positive integer $N$ is a [i]palindrome[/i] if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following two properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.
How many integers between $80$ and $100$ are prime?
$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$
We are given a semicircle $k$ with center $O$ and diameter $AB$. Let $C$ be a point on $k$ such that $CO \bot AB$. The bisector of $\angle ABC$ intersects $k$ at point $D$. Let $E$ be a point on $AB$ such that $DE \bot AB$ and let $F$ be the midpoint of $CB$. Prove that the quadrilateral $EFCD$ is cyclic.
Prove that every convex polygon has at most one triangulation consisting entirely of acute triangles.
Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point.
[i]Proposed by Brandon Wang[/i]
How many trailing zeros does the value
\[300\cdot305\cdot310\dots1090\cdot1095\cdot1100\]
end with?
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality
$$|f(x + y) + \sin x + \sin y| < 2?$$
For every odd number $p>1$ we have:
$\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad
\textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\
\textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad
\textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\
\textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]
Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
[i]Proposed by Andrew Wu[/i]
Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Aditya Rao[/i]
On the trip home from the meeting where this AMC$ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties?
$ \textbf{(A)}\ 12\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 18\qquad
\textbf{(D)}\ 20\qquad
\textbf{(E)}\ 24$
A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?
Find all primes $p$ such that the expression
\[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\]
is divisible by $p^3$.
Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$
On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?
[asy]
draw(Circle(origin, 2));
draw(Circle(origin, 1));
draw(origin--2*dir(90));
draw(origin--2*dir(210));
draw(origin--2*dir(330));
label("$1$", 0.35*dir(150), dir(150));
label("$1$", 1.3*dir(30), dir(30));
label("$1$", (0,-1.3), dir(270));
label("$2$", 1.3*dir(150), dir(150));
label("$2$", 0.35*dir(30), dir(30));
label("$2$", (0,-0.35), dir(270));[/asy]
$ \textbf{(A)}\: \frac{17}{36}\qquad \textbf{(B)}\: \frac{35}{72}\qquad \textbf{(C)}\: \frac{1}{2}\qquad \textbf{(D)}\: \frac{37}{72}\qquad \textbf{(E)}\: \frac{19}{36}\qquad $
The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when:
a) $k=3$
b) $k=4$
c) $k=5$ .