Found problems: 85335
2011 AMC 10, 13
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} $
2015 AoPS Mathematical Olympiad, 3
A small apartment building has four doors, with door numbers $1, 2, 3, 4.$ John has $2^4-1=15$ keys, label with of possible nonempty subsets of $\{1,2,3,4\}$, but he forgot which key is which. If an element on the key matches the door number, the key can open the door (e.g. key $\{1,2,4\}$ can open Door 4). He picks a key at random and tries to open Door 1, which fails, so he discards it. John then randomly picks one of his remaining 14 keys and tries to open Door 2, but it doesn't open, so he throws away that key as well. He then randomly selects one of the remaining 13 keys, and tests it on Door 3. What is the probability that it will open?
[i]Proposed by dantx5[/i]
2017 IFYM, Sozopol, 5
In a group of $n$ people $A_1,A_2… A_n$ each one has a different height. On each turn we can choose any three of them and figure out which one of them is the highest and which one is the shortest. What’s the least number of turns one has to make in order to arrange these people by height, if:
a) $n=5$; b) $n=6$; c) $n=7$?
2001 Iran MO (2nd round), 2
Let $ABC$ be an acute triangle. We draw $3$ triangles $B'AC,C'AB,A'BC$ on the sides of $\Delta ABC$ at the out sides such that:
\[ \angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} \ \ \ , \ \ \ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ} \]
If $M$ is the midpoint of side $BC$, prove that $B'M$ is perpendicular to $A'C'$.
2006 AMC 12/AHSME, 14
Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
$ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$
2015 China Team Selection Test, 2
Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]
1993 Baltic Way, 19
A convex quadrangle $ ABCD$ is inscribed in a circle with center $ O$. The angles $ AOB, BOC, COD$ and $ DOA$, taken in some order, are of the same size as the angles of the quadrangle $ ABCD$.
Prove that $ ABCD$ is a square
2007 AIME Problems, 14
Let a sequence be defined as follows: $a_{1}= 3$, $a_{2}= 3$, and for $n \ge 2$, $a_{n+1}a_{n-1}= a_{n}^{2}+2007$. Find the largest integer less than or equal to $\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}$.
2021 Romania National Olympiad, 1
Let $f:[a,b] \rightarrow \mathbb{R}$ a function with Intermediate Value property such that $f(a) * f(b) < 0$. Show that there exist $\alpha$, $\beta$ such that $a < \alpha < \beta < b$ and $f(\alpha) + f(\beta) = f(\alpha) * f(\beta)$.
2011 VTRMC, Problem 7
Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.
2024 Sharygin Geometry Olympiad, 9.2
Points $A, B, C, D$ on the plane do not form a rectangle. Let the sidelengths of triangle $T$ equal $AB+CD$, $AC+BD$, $AD+BC$. Prove that the triangle $T$ is acute-angled.
1997 VJIMC, Problem 3
Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$:
$$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)
1992 Czech And Slovak Olympiad IIIA, 1
For a permutation $p(a_1,a_2,...,a_{17})$ of $1,2,...,17$, let $k_p$ denote the largest $k$ for which $a_1 +...+a_k < a_{k+1} +...+a_{17}$. Find the maximum and minimum values of $k_p$ and find the sum $\sum_{p} k_p$ over all permutations$ p$.
2023 MOAA, 15
Triangle $ABC$ has $AB = 5$, $BC = 7$, $CA = 8$. Let $M$ be the midpoint of $BC$ and let points $P$ and $Q$ lie on $AB$ and $AC$ respectively such that $MP \perp AB$ and $MQ \perp AC$. If $H$ is the orthocenter of $\triangle{APQ}$ then the area of $\triangle{HPM}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers and $b$ is square-free. Find $a+b+c$.
[i]Proposed by Harry Kim[/i]
2016 Fall CHMMC, 14
For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.
2022 AMC 10, 15
Let $S_n$ be the sum of the first $n$ term of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$?
$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$
1980 Polish MO Finals, 2
Prove that for every $n$ there exists a solution of the equation
$$a^2 +b^2 +c^2 = 3abc$$
in natural numbers $a,b,c$ greater than $n$.
2004 Iran MO (3rd Round), 14
We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n)$.
a) Show that if $ \gcd(m,n)\equal{}1$ then we have $ f(mn)\equal{}f(m)\cdot f(n)$;
b) Show that $ \sum_{d|n}f(d) \equal{} nd(n)$.
1989 Cono Sur Olympiad, 2
Find the sum\[1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.\]
2017 Purple Comet Problems, 5
A store had $376$ chocolate bars. Min bought some of the bars, and Max bought $41$ more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.
2002 Tournament Of Towns, 3
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
1987 IMO Shortlist, 19
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than
\[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\]
[i]Proposed by Soviet Union[/i]
III Soros Olympiad 1996 - 97 (Russia), 10.2
It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.
2021 Romanian Master of Mathematics, 4
Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves.
[i]Proposed by China[/i]
2004 Estonia National Olympiad, 1
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$