Found problems: 85335
2001 Swedish Mathematical Competition, 5
Find all polynomials $p(x)$ such that $p'(x)^2 = c p(x) p''(x)$ for some constant $c$.
LMT Guts Rounds, 2020 F5
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8?$
[i]Proposed by Alex Li[/i]
2004 Poland - Second Round, 2
In convex hexagon $ ABCDEF$ all sides have equal length and
$ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$.
Prove that the diagonals $ AD,BE,CF$ are concurrent.
2018 CMIMC Algebra, 7
Compute
\[\sum_{k=0}^{2017}\dfrac{5+\cos\left(\frac{\pi k}{1009}\right)}{26+10\cos\left(\frac{\pi k}{1009}\right)}.\]
1978 Bulgaria National Olympiad, Problem 5
Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.
[i]Jordan Tabov[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.1
A car drove from one city to another. She drove the first third of the journey at a speed of $50$ km/h, the second third at $60$ km/h, and the last third at $70$ km/h. What is the average speed of the car along the entire journey?
2024 JHMT HS, 5
Compute the positive difference between the two solutions to the equation $2x^2-28x+9=0$.
2003 China Team Selection Test, 3
Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that:
$\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$
2016 Azerbaijan JBMO TST, 1
If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality:
$$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$
2020 USMCA, 11
A permutation of $USMCAUSMCA$ is selected uniformly at random. What is the probability that this permutation is exactly one transposition away from $USMCAUSMCA$ (i.e. does not equal $USMCAUSMCA$, but can be turned into $USMCAUSMCA$ by swapping one pair of letters)?
2014 Purple Comet Problems, 3
The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy]
2009 Balkan MO Shortlist, A2
Let $ABCD$ be a square and points $M$ $\in$ $BC$, $N \in CD$, $P$ $\in$ $DA$, such that $\angle BAM$ $=$ $x$, $\angle CMN$ $=$ $2x$, $\angle DNP$ $=$ $3x$
[list=i]
[*] Show that, for any $x \in (0, \tfrac{\pi}{8} )$, such a configuration exists
[*] Determine the number of angles $x \in ( 0, \tfrac{\pi}{8} )$ for which $\angle APB =4x$
2004 Bulgaria Team Selection Test, 3
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
2002 Moldova National Olympiad, 2
For every nonnegative integer $ n$ and every real number $ x$ prove the inequality:
$ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
2002 Romania Team Selection Test, 1
Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$.
Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number.
[i]Laurentiu Panaitopol[/i]
2008 ITest, 69
In the sequence in the previous problem, how many of $u_1,u_2,u_3,\ldots, u_{2008}$ are pentagonal numbers?
2007 Harvard-MIT Mathematics Tournament, 3
Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$.
2009 Regional Olympiad of Mexico Center Zone, 3
An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of length $ 1$, drawing parallel lines (at distance $ 1$) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
Kyiv City MO Juniors 2003+ geometry, 2018.7.41
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
2004 AMC 10, 2
For any three real numbers $ a$, $ b$, and $ c$, with $ b\neq c$, the operation $ \otimes$ is defined by:
\[ \otimes(a,b,c) \equal{} \frac {a}{b \minus{} c}
\]What is $ \otimes(\otimes(1,2,3), \otimes(2,3,1),\otimes(3,1,2))$?
$ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(B)}\ \minus{}\!\frac {1}{4}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac {1}{4}\qquad
\textbf{(E)}\ \frac {1}{2}$
2023 Indonesia TST, 1
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2008 Peru IMO TST, 1
Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way.
Prove that $ La$ $ Lb$ and $ Lc$ are concurrent.
Daniel
1995 AIME Problems, 15
Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2014 Contests, 3
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.