Found problems: 85335
2016 NIMO Problems, 2
A time is chosen randomly and uniformly in an 24-hour day. The probability that at that time, the (non-reflex) angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$.
[i]Proposed by Yannick Yao[/i]
2017 Thailand TSTST, 3
In $\vartriangle ABC$ with $AB > AC$, the tangent to the circumcircle at $A$ intersects line $BC$ at $P$. Let $Q$ be the point on $AB$ such that $AQ = AC$, and $A$ lies between $B$ and $Q$. Let $R$ be the point on ray $AP$ such that $AR = CP$. Let $X, Y$ be the midpoints of $AP, CQ$ respectively. Prove that $CR = 2XY$ .
2010 BMO TST, 4
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.
2005 National Olympiad First Round, 30
Which of the following cannot be the number of positive integer divisors of the number $n^2+1$, where $n$ is an integer?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of above}
$
1998 National Olympiad First Round, 24
Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is
$\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ \left\{0,3,4,6\right\} \\ \textbf{(D)}\ \left\{0,2,3,4,6\right\} \\ \textbf{(E)}\ \left\{0,2,3,4\right\}$
2019 China Western Mathematical Olympiad, 6
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $a_1\leq a_2 \leq \cdots \leq a_n .$ Prove that
$$\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.$$
2015 Romanian Master of Mathematics, 5
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
2013 Saudi Arabia BMO TST, 3
Solve the following equation where $x$ is a real number: $\lfloor x^2 \rfloor -10\lfloor x \rfloor + 24 = 0$
2009 Germany Team Selection Test, 1
Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.
Russian TST 2020, P2
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
2016 Korea Junior Math Olympiad, 6
circle $O_1$ is tangent to $AC$, $BC$(side of triangle $ABC$) at point $D, E$.
circle $O_2$ include $O_1$, is tangent to $BC$, $AB$(side of triangle $ABC$) at point $E, F$
The tangent of $O_2$ at $P(DE \cap O_2, P \neq E)$ meets $AB$ at $Q$.
A line passing through $O_1$(center of $O_1$) and parallel to $BO_2$($O_2$ is also center of $O_2$) meets $BC$ at $G$, $EQ \cap AC=K, KG \cap EF=L$, $EO_2$ meets circle $O_2$ at $N(\neq E)$, $LO_2 \cap FN=M$.
IF $N$ is a middle point of $FM$, prove that $BG=2EG$
2008 IberoAmerican, 1
The integers from 1 to $ 2008^2$ are written on each square of a $ 2008 \times 2008$ board. For every row and column the difference between the maximum and minimum numbers is computed. Let $ S$ be the sum of these 4016 numbers. Find the greatest possible value of $ S$.
2017 JBMO Shortlist, G3
Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$ . Let $S$ be the midpoint of segment $AD$. If $\angle ADE = \angle ABC - \angle ACB$ prove that $\angle BSC = 2 \angle BAC$ .
Today's calculation of integrals, 870
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$
(1) Find all points of intersection of $E$ and $H$.
(2) Find the area of the region expressed by the system of inequality
\[\left\{
\begin{array}{ll}
3x^2+y^2\leq 3 &\quad \\
xy\geq \frac 34 , &\quad
\end{array}
\right.\]
2000 All-Russian Olympiad Regional Round, 10.5
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality
$$|f(x + y) + \sin x + \sin y| < 2?$$
1998 Estonia National Olympiad, 3
The hotel has $13$ rooms with rooms from $1$ to $13$, located on one side of a straight corridor in ascending order of numbers. During the tourist season, which lasts from May $1$st to October $1$st, the hotel visitor has the opportunity to rent either one room for two days in a row, or two adjacent rooms together for one day. How much could a hotel owner earn in a season if it is known that on October $1$, rooms $1$ and $13$ were empty, and the payment for one room was one tugrik per day?
1970 IMO Longlists, 36
Let $x, y, z$ be non-negative real numbers satisfying
\[x^2 + y^2 + z^2 = 5 \quad \text{ and } \quad yz + zx + xy = 2.\]
Which values can the greatest of the numbers $x^2 -yz, y^2 - xz$ and $z^2 - xy$ have?
2011 AMC 12/AHSME, 9
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{4}{9} \qquad
\textbf{(D)}\ \frac{5}{9} \qquad
\textbf{(E)}\ \frac{2}{3} $
2014 PUMaC Number Theory A, 3
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
1999 Romania National Olympiad, 1
Let $AD$ be the bisector of angle $A$ of the triangle $ABC$. One considers the points M, N on the half-lines $(AB$ and $(AC$, respectively, such that $\angle MDA = \angle B$ and $\angle NDA = \angle C$. Let $AD \cap MN=\{P\}$. Prove that: $$AD^3 = AB \cdot AC\cdot AP$$
2019 Romania Team Selection Test, 4
Let be two natural numbers $ m,n, $ and $ m $ pairwise disjoint sets of natural numbers $ A_0,A_1,\ldots ,A_{m-1}, $ each having $ n $ elements, such that no element of $ A_{i\pmod m} $ is divisible by an element of $ A_{i+1\pmod m} , $ for any natural number $ i. $
Determine the number of ordered pairs
$$ (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j $$
such that $ a|b $ and such that $ \{ a,b \}\not\in A_k, $ for any $ k\in\{ 0,1,\ldots ,m-1 \} . $
[i]Radu Bumbăcea[/i]
2010 Peru IMO TST, 6
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.
[i]Proposed by Eugene Bilopitov, Ukraine[/i]
2020 Princeton University Math Competition, A5/B7
Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $AD$, $BE$, $CF$ all contain a point $K$. Let $L$ be the second intersection of the circumcircles of $BFK$ and $CEK$. Suppose that $\frac{AK}{KD} = \frac{11}{7}$ , and $BD = 6$. If $KL^2 =\frac{a}{b}$, where $a, b$ are relatively prime integers, find $a + b$.
2018 PUMaC Combinatorics B, 5
Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.
2012 Saint Petersburg Mathematical Olympiad, 3
At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC * AD = BD * AC$. Also $ \angle ADS =\angle BDS ,\angle ACS =\angle BCS$.
Prove that the plane $SAB$ is perpendicular to the plane of the base.