Found problems: 85335
Points on complex plane that complex numbers $z_1,z_2$ corresponding to are $A,B$, and $|z_1|=4,4z_1^2-2z_1z_2+z_2^2=0$. $O$ is original point, then the area of $\triangle OAB$ is
$\text{(A)}8\sqrt3\qquad\text{(B)}4\sqrt3\qquad\text{(C)}6\sqrt3\qquad\text{(D)}12\sqrt3$
Show that there are $1977$ non-similar triangles such that the angles $A, B, C$ satisfy $\frac{\sin A + \sin B + \sin C}{\cos A +\cos B + \cos C} = \frac{12}{7}$ and $\sin A \sin B \sin C = \frac{12}{25}$.
Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$?
[i]Proposed by Oleksiy Masalitin[/i]
Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$
Prove that
$a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$
Show that in a non-equilateral triangle, the following statements are equivalent:
$(a)$ The angles of the triangle are in arithmetic progression.
$(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.
At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
On a plane, $n$ straight lines are drawn. Two domains are called [i]adjacent [/i] if they border by a line segment. Prove that the domains into which the plane is divided by these lines can be painted two colors so that no two [i]adjacent [/i] domains are of the same color.
Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely?
[i](Proposed by Wong Jer Ren)[/i]
On this monthly calendar, the date behind one of the letters is added to the date behind $C$. If this sum equals the sum of the dates behind $A$ and $B$, then the letter is
[asy]
unitsize(12);
draw((1,1)--(23,1));
draw((0,5)--(23,5));
draw((0,9)--(23,9));
draw((0,13)--(23,13));
for(int a=0; a<6; ++a)
{
draw((4a+2,0)--(4a+2,14));
}
label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N);
label("Fri.",(16,14),N); label("Sat.",(20,14),N);
label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N);
label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N);
label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy]
$ \text{(A)}\ \text{P}\qquad\text{(B)}\ \text{Q}\qquad\text{(C)}\ \text{R}\qquad\text{(D)}\ \text{S}\qquad\text{(E)}\ \text{T} $
Divide the set of twelve numbers $A = \{3, 4, 5, ...,13, 14\}$ into two sets $ B$ and $C$ 'of six numbers each according to this condition: for any two different numbers with $ B$ their sum does not belong to $ B$ and for any two different numbers from $C$, the sum does not belong to $C$.
Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
$\text{(A)}\ \frac{1}{10} \qquad \text{(B)}\ \frac{1}{4} \qquad \text{(C)}\ \frac{1}{3} \qquad \text{(D)}\ \frac{2}{5} \qquad \text{(E)}\ \frac{1}{2}$
Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$.
[i]Proposed by Andrew Wu[/i]
(Note: wording changed from original to specify that $a \neq 20, 22$.)
Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression of positive real numbers, and $ m $ be a natural number. Calculate:
[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $
[b]b)[/b] $ \lim_{n\to\infty } \frac{1}{a_n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $
[i]Dumitru Acu[/i]
The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?
Suppose $S$ is the set of all positive integers. For $a,b \in S$, define
\[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\]
For example $8*12=6$.
Show that [b]exactly two[/b] of the following three properties are satisfied:
(i) If $a,b \in S$, then $a*b \in S$.
(ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$.
(iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
$ \textbf{(A)}\ 41\qquad
\textbf{(B)}\ 42\qquad
\textbf{(C)}\ 43\qquad
\textbf{(D)}\ 44\qquad
\textbf{(E)}\ 45$
For positive integers $p$, $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls.
We now want to construct a cuboid of side lengths $p$, $q$ and $r$ out of the cubes, without tearing the thread.
a) For which numbers $p$, $q$ and $r$ is this possible?
b) For which numbers $p$, $q$ and $r$ is this possible in a way such that both ends of the thread coincide?
An geometric progression starting at $a_0 = 3$ has an even number of terms. Suppose the difference between the odd indexed terms and even indexed terms is $39321$ and that the sum of the fi rst and last term is $49155$. Find the common ratio of this geometric progression.
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that
$ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$
Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$.
Prove that:
\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0\]
[i]Proposed by Nikolay Nikolov[/i]
Let $F_1, F_2, ...$ be Borel-measurable sets on the plane whose union is the whole plane. Prove that there is a natural number n and circle S for which the set $S \cap F_n$ is dense in S. Also show that the statement is not necessarily true if we omit the condition for the measurability of sets $F_j$.
A point $ P$ is selected at random from the interior of the pentagon with vertices $ A \equal{} (0,2)$, $B \equal{} (4,0)$, $C \equal{} (2 \pi \plus{} 1, 0)$, $D \equal{} (2 \pi \plus{} 1,4)$, and $ E \equal{} (0,4)$. What is the probability that $ \angle APB$ is obtuse?
[asy]
size(150);
pair A, B, C, D, E;
A = (0,1.5);
B = (3,0);
C = (2 *pi + 1, 0);
D = (2 * pi + 1,4);
E = (0,4);
draw(A--B--C--D--E--cycle);
label("$A$", A, dir(180));
label("$B$", B, dir(270));
label("$C$", C, dir(0));
label("$D$", D, dir(0));
label("$E$", E, dir(180));
[/asy]
$ \displaystyle \textbf{(A)} \ \frac {1}{5} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)} \ \frac {5}{16} \qquad \textbf{(D)} \ \frac {3}{8} \qquad \textbf{(E)} \ \frac {1}{2}$
We consider triangles $ABC$, in which the point $M$ lies on the side $AB$, $AM = a$, $BM = b$, $CM = c$ ($c <a, c <b$). Find the smallest radius of the circumcircle of such triangles.
Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$
[i]Proposed by Henry Jiang[/i]