Found problems: 85335
Estonia Open Junior - geometry, 1996.2.4
A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.
Brazil L2 Finals (OBM) - geometry, 2016.4
Consider a scalene triangle $ ABC $ with $ AB <AC <BC. $ The $ AB $ side mediator cuts the $ B $ side at the $ K $ point and the $ AC $ prolongation at the $ U. $ point. $ AC $ side cuts $ BC $ side at $ O $ point and $ AB $ side extension at $ G$ point. Prove that the $ GOKU $ quad is cyclic, meaning its four vertices are at same circumference
2010 Slovenia National Olympiad, 2
Find all real $x$ in the interval $[0, 2\pi)$ such that
\[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\]
2002 India IMO Training Camp, 21
Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
2011 AMC 10, 22
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$ \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750 $
2022 Philippine MO, 8
The set $S = \{1, 2, \dots, 2022\}$ is to be partitioned into $n$ disjoint subsets $S_1, S_2, \dots, S_n$ such that for each $i \in \{1, 2, \dots, n\}$, exactly one of the following statements is true:
(a) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) > 1.$
(b) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) = 1.$
Find the smallest value of $n$ for which this is possible.
2019 Sharygin Geometry Olympiad, 5
Grade8 P5 of Sharygin 2019 Finals
2004 Harvard-MIT Mathematics Tournament, 9
A classroom consists of a $5\times5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
2004 AIME Problems, 15
A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?
2003 Poland - Second Round, 4
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.
2006 MOP Homework, 5
Smallville is populated by unmarried men and women, some of which are acquainted. The two City Matchmakers know who is acquainted with whom. One day, one of the matchmakers claimed: "I can arrange it so that every red haired man will marry a woman with who he is acquainted." The other matchmaker claimed: "I can arrange it so that every blonde woman will marry a man with whom she is acquainted." An amateur mathematician overheard this conversation and said: "Then it can be arranged so that every red haired man will marry a woman with whom he is acquainted and at the same time very blonde woman will marry a man with who she is acquainted." Is the mathematician right?
2018 Purple Comet Problems, 28
In $\vartriangle ABC$ points $D, E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF,$ fi nd the side length $BC$.
1979 Kurschak Competition, 1
The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.
2020 LMT Fall, 35
Estimate the number of ordered pairs $(p,q)$ of positive integers at most $2020$ such that the cubic equation $x^3-px-q=0$ has three distinct real roots. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\Big\lfloor15\min\Big(\frac{A}{E},\frac{E}{A}\Big)\Big\rfloor.\]
[i]Proposed by Alex Li[/i]
2008 National Olympiad First Round, 2
For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers?
$
\textbf{(A)}\ 301
\qquad\textbf{(B)}\ 403
\qquad\textbf{(C)}\ 415
\qquad\textbf{(D)}\ 427
\qquad\textbf{(E)}\ 481
$
2006 Rioplatense Mathematical Olympiad, Level 3, 1
The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.
2016 Bundeswettbewerb Mathematik, 2
A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area.
What is the area of triangle $XYZ$ at the end of the game, if both play optimally?
2024 Bulgarian Winter Tournament, 9.2
Let $p>q$ be primes, such that $240 \nmid p^4-q^4$. Find the maximal value of $\frac{q} {p}$.
2012 Online Math Open Problems, 2
How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once?
[i]Author: Ray Li[/i]
2014 Austria Beginners' Competition, 1
Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$.
(W. Janous, Innsbruck)
2016 Romania National Olympiad, 1
Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $
Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $
2018 CCA Math Bonanza, I2
Let $P$ be the product of the first $50$ nonzero square numbers. Find the largest integer $k$ such that $7^k$ divides $P$.
[i]2018 CCA Math Bonanza Individual Round #2[/i]
2011 All-Russian Olympiad, 2
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
LMT Speed Rounds, 2010.18
Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$