Found problems: 329
2010 Contests, 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.
1995 AMC 8, 8
An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = $ \$ 1.60$, how much lire will the traveler receive in exchange for $ \$ 1.00$?
$\text{(A)}\ 180 \qquad \text{(B)}\ 480 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 1875 \qquad \text{(E)}\ 4875$
2011 Math Prize For Girls Problems, 7
If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]
what is the value of
\[
z^{2010} + z^{-2010} \, ?
\]
2008 Turkey Team Selection Test, 2
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.
2005 Vietnam National Olympiad, 2
Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:
a) \[ CD\leq R \]
b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).
2012 Brazil Team Selection Test, 5
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
2012 Today's Calculation Of Integral, 818
For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$
2010 AMC 12/AHSME, 18
A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step?
$ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$
1981 AMC 12/AHSME, 14
In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is
$\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$
1996 Polish MO Finals, 3
$a_i, x_i$ are positive reals such that $a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1$. Show that
\[ 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} \]
When do we have equality?
2004 Iran MO (2nd round), 1
$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that:
\[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]
2013 AMC 10, 2
Alice is making a batch of cookies and needs $2 \frac{1}{2}$ cups of sugar. Unforunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ 20$
2011 AIME Problems, 2
In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$, $DF=8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m,n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.
2012 AMC 12/AHSME, 24
Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by $$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$ and more generally \[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \]
Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$?
$ \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 $
1990 APMO, 2
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
1986 AMC 12/AHSME, 3
$\triangle ABC$ is a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ is the bisector of $\angle ABC$, then $\angle BDC =$
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A= origin, B = 3 * dir(25), C = (B.x,0);
pair X = bisectorpoint(A,B,C), D = extension(B,X,A,C);
draw(B--A--C--B--D^^rightanglemark(A,C,B,4));
path g = anglemark(A,B,D,14);
path h = anglemark(D,B,C,14);
draw(g);
draw(h);
add(pathticks(g,1,0.11,6,6));
add(pathticks(h,1,0.11,6,6));
label("$A$",A,W);
label("$B$",B,NE);
label("$C$",C,E);
label("$D$",D,S);
label("$20^\circ$",A,8*dir(12.5));
[/asy]
$ \textbf{(A)}\ 40^\circ \qquad
\textbf{(B)}\ 45^\circ \qquad
\textbf{(C)}\ 50^\circ \qquad
\textbf{(D)}\ 55^\circ \qquad
\textbf{(E)}\ 60^\circ $
2010 Iran MO (3rd Round), 4
in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)
2006 Romania Team Selection Test, 1
Let $r$ and $s$ be two rational numbers. Find all functions $f: \mathbb Q \to \mathbb Q$ such that for all $x,y\in\mathbb Q$ we have \[ f(x+f(y)) = f(x+r)+y+s. \]
2020 BAMO, D/2
Here’s a screenshot of the problem. If someone could LaTEX a diagram, that would be great!
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.
1991 Baltic Way, 10
Express the value of $\sin 3^\circ$ in radicals.
2007 IMC, 6
How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length?
2009 Iran Team Selection Test, 12
$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that :
$ |T| \leq \frac {4}{9}n + \log_2 n + 2$
2014 Math Prize For Girls Problems, 16
If $\sin x + \sin y = \frac{96}{65}$ and $\cos x + \cos y = \frac{72}{65}$, then what is the value of $\tan x + \tan y$?
1959 AMC 12/AHSME, 44
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$
$ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $
$\textbf{(E)}\text{ must be between -1 and +1} $