Found problems: 85335
A lake is in the shape of a regular hexagon of side length \(1\). Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance \(\sqrt{3}\) (measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have \(2025\) leaves in the lake?
The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
Problems 17, 18, and 19 refer to the following:
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.
The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)
$ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
Let $\alpha$ be a complex number such that both $\alpha$ and $\alpha+1$ have modulus $1$. If for a positive integer $n$, $1+\alpha$ is an $n$-th root of unity, then show that $\alpha$ is also an $n$-th root of unity and $n$ is a multiple of $6$.
Let three circles $ \Gamma_1, \Gamma_2, \Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $ P$ in the plane, outside the three circles, construct six points $ A_1, B_1, A_2, B_2, A_3, B_3$ as follows: For each $ i \equal{} 1, 2, 3$, $ A_i, B_i$ are distinct points on the circle $ \Gamma_i$ such that the lines $ PA_i$ and $ PB_i$ are both tangents to $ \Gamma_i$. Call the point $ P$ exceptional if, from the construction, three lines $ A_1B_1, A_2 B_2, A_3 B_3$ are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.
Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations:
$$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$
Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as
$$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$
doesn't have limit at $ \infty . $
[i]Romeo Ilie[/i]
What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?
Solve the equation $$ \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.$$
Let $ABC$ be a triangle with side lengths $AB=6, BC=7, CA=8$ and circumcircle $\omega.$ Denote $M$ to be the midpoint of $BC.$ Let $P$ be the intersection of the tangent to $\omega$ at $A$ and $BC.$ The line parallel to $BC$ passing through $A$ intersects $\omega$ at another point $D.$ The tangent to $\omega$ passing through $P$ that is not $PA$ intersects $DM$ at a point $Q.$ Denote $J$ to be the intersection of $(BMQ)$ and $AQ.$ Extend $BJ$ to intersect $AC$ at $E.$ Compute $\tfrac{BJ}{JE}.$
[i]Lightning 4.4[/i]
$ABCD$ is a cyclic quadrilateral in which $AB=4$, $BC=3$, $CD=2$, and $AD=5$. Diagonals $AC$ and $BD$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $BD$ at $X$. $\omega$ intersects $AB$ and $AD$ at $Y$ and $Z$ respectively. Compute $YZ/BD$.
A bowl contained $10\%$ blue candies and $25\%$ red candies. A bag containing three quarters red candies and one quarter blue candies was added to the bowl. Now the bowl is $16\%$ blue candies. What percentage of the candies in the bowl are now red?
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$.
(N Vassiliev)
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \sqrt2 \qquad
\textbf{(C)}\ \sqrt3 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 2\sqrt2$
The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.
Let $a_1,...,a_n$ be $n$ real numbers. If for each odd positive integer $k\leqslant n$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$, then for each odd positive integer $k$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$.
[i]Proposed by M. Didin[/i]
Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.
Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$.
[i]S. Berlov, F. Petrov, A. Akopyan[/i]
Mike has two similar pentagons. The first pentagon has a perimeter of $18$ and an area of $8 \frac{7}{16}$ . The second pentagon has a perimeter of $24$. Find the area of the second pentagon.
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.
[i]Robin Son[/i]
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.
Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$
for all $z\in \mathbb{C}$.
Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $