Found problems: 329
2008 AMC 12/AHSME, 8
Points $ B$ and $ C$ lie on $ \overline{AD}$. The length of $ \overline{AB}$ is $ 4$ times the length of $ \overline{BD}$, and the length of $ \overline{AC}$ is $ 9$ times the length of $ \overline{CD}$. The length of $ \overline{BC}$ is what fraction of the length of $ \overline{AD}$?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{13} \qquad
\textbf{(C)}\ \frac{1}{10} \qquad
\textbf{(D)}\ \frac{5}{36} \qquad
\textbf{(E)}\ \frac{1}{5}$
2005 MOP Homework, 4
Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]
2011 Balkan MO, 4
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.
2007 Regional Competition For Advanced Students, 3
Let $ a$ be a positive real number and $ n$ a non-negative integer. Determine $ S\minus{}T$, where
$ S\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{(k\minus{}1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$ and $ T\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$
2012 Turkey Team Selection Test, 3
Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?
2007 Iran MO (3rd Round), 2
$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]
2012 AMC 12/AHSME, 13
Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?
$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $
2008 Harvard-MIT Mathematics Tournament, 8
Trodgor the dragon is burning down a village consisting of $ 90$ cottages. At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?
2009 Today's Calculation Of Integral, 438
Evaluate $ \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.$
1988 IMO Longlists, 90
Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]
2004 National Olympiad First Round, 17
Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$
$
\textbf{(A)}\ 2\sqrt 3
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \dfrac 13
\qquad\textbf{(D)}\ \dfrac 12
\qquad\textbf{(E)}\ 1
$
1971 AMC 12/AHSME, 33
If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S$, $S'$, and $n$ is
$\textbf{(A) }(SS')^{\frac{1}{2}n}\qquad\textbf{(B) }(S/S')^{\frac{1}{2}n}\qquad\textbf{(C) }(SS')^{n-2}\qquad\textbf{(D) }(S/S')^n\qquad \textbf{(E) }(S/S')^{\frac{1}{2}(n-1)}$
1995 India National Olympiad, 5
Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]
2008 Germany Team Selection Test, 1
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$
2004 AIME Problems, 6
An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
2010 India IMO Training Camp, 10
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
2007 AMC 8, 6
The average cost of a long-distance call in the USA in $1985$ was $41$ cents per minute, and the average cost of a long-distance call in the USA in $2005$ was $7$ cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
$\textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 17 \qquad
\textbf{(C)}\ 34 \qquad
\textbf{(D)}\ 41 \qquad
\textbf{(E)}\ 80$
2014 Bosnia Herzegovina Team Selection Test, 2
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
2014 AIME Problems, 6
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
2005 Balkan MO, 3
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
1977 AMC 12/AHSME, 4
[asy]
size(130);
pair A = (2, 2.4), C = (0, 0), B = (4.3, 0),
E = 0.7*A, F = 0.57*A + 0.43*B, D = (2.4, 0);
draw(A--B--C--cycle);
draw(E--D--F);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C, W);
label("$D$", D, S);
label("$E$", E, NW);
label("$F$", F, NE);
//Credit to MSTang for the diagram[/asy]
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
$\textbf{(A) }30^\circ\qquad\textbf{(B) }40^\circ\qquad\textbf{(C) }50^\circ\qquad\textbf{(D) }65^\circ\qquad \textbf{(E) }\text{none of these}$
2016 Indonesia TST, 4
We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set
\[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).
[i]Proposed by Javad Abedi[/i]
2013 Princeton University Math Competition, 6
Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.
2011 India IMO Training Camp, 3
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
2015 AMC 12/AHSME, 15
At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?
$\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$