Found problems: 85335
CIME II 2018, 6
Define $f(x)=-\frac{2x}{4x+3}$ and $g(x)=\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x)=g(f(h^n(x)))$, where $h^1(x)=g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$ can be written in the form $ab^c$, for some integers $a,b,c$ where $c$ is as maximal as possible and $b\ne 1$, find $a+b+c$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2018 Purple Comet Problems, 3
The fraction
$$\left(\frac{\frac13+1}{3} +\frac{1+ \frac13}{3} \right) / \left(\frac{3}{\frac{1}{3+1}+\frac{ 1}{1+3}}\right)$$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1999 Ukraine Team Selection Test, 2
Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.
2019 Novosibirsk Oral Olympiad in Geometry, 3
The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square.
(Distance in the picture is measured horizontally from the midpoint of the side of the square.)
[img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]
2014 AMC 8, 9
In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
[asy]
size(300);
defaultpen(linewidth(0.8));
pair A=(-1,0),C=(1,0),B=dir(40),D=origin;
draw(A--B--C--A);
draw(D--B);
dot("$A$", A, SW);
dot("$B$", B, NE);
dot("$C$", C, SE);
dot("$D$", D, S);
label("$70^\circ$",C,2*dir(180-35));
[/asy]
$\textbf{(A) }100\qquad\textbf{(B) }120\qquad\textbf{(C) }135\qquad\textbf{(D) }140\qquad \textbf{(E) }150$
2023 Oral Moscow Geometry Olympiad, 3
Given is a triangle $ABC$ and $M$ is the midpoint of the minor arc $BC$. Let $M_1$ be the reflection of $M$ with respect to side $BC$. Prove that the nine-point circle bisects $AM_1$.
2024 IFYM, Sozopol, 5
Depending on the real number \( a \), find all polynomials \( P(x) \) with real coefficients such that
\[
(x^3 - ax^2 + 1)P(x) = (x^3 + ax^2 + 1)P(x-1)
\]
for every real number \( x \).
2011 ELMO Shortlist, 3
Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples.
[i]Mitchell Lee.[/i]
2009 Stanford Mathematics Tournament, 11
Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$. Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$.
1981 All Soviet Union Mathematical Olympiad, 323
The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?
2015 239 Open Mathematical Olympiad, 3
Positive integers are colored either blue or red such that if $a,b$ have the same color and $a-10b$ is a positive integer then $a-10b, a$ have the same color as well. How many such coloring exist?
2012 AMC 8, 1
Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}6\dfrac23 \qquad \textbf{(C)}\hspace{.05in}7\dfrac12 \qquad \textbf{(D)}\hspace{.05in}8 \qquad \textbf{(E)}\hspace{.05in}9 $
2008 China Western Mathematical Olympiad, 4
Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 \equal{} b$ and $ x_{n \plus{} 1} \equal{} ax^{m}_{n} \plus{} b$, $ n \equal{} 1,2,...$. Prove that
(1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq \minus{} 2$;
(2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m \minus{} 1}\geq\frac {(m \minus{} 1)^{m \minus{} 1}}{m^m}$.
2017 Regional Olympiad of Mexico Southeast, 6
Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.
1957 AMC 12/AHSME, 22
If $ \sqrt{x \minus{} 1} \minus{} \sqrt{x \plus{} 1} \plus{} 1 \equal{} 0$, then $ 4x$ equals:
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 4\sqrt{\minus{}1}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 1\frac{1}{4}\qquad
\textbf{(E)}\ \text{no real value}$
2003 Gheorghe Vranceanu, 1
Prove that if a $ 2\times 2 $ complex matrix has the property that there exists a natural number $ n $ such that $ \text{tr}\left( A^n\right) =\text{tr}\left( A^{n+1} \right) =0, $ then $ A^2=0. $
1963 German National Olympiad, 6
Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals.
a) Prove that $BCPN$ is also a cyclic quadrilateral.
b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.
2010 Nordic, 2
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that
\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]
2000 Moldova National Olympiad, Problem 2
Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.
2001 Moldova National Olympiad, Problem 3
For an arbitrary point $D$ on side $BC$ of an acute-angled triangle $ABC$, let $O_1$ and $O_2$ be the circumcenters of the triangles $ABD$ and $ACD$, and $O$ be the circumcenter of the triangle $AO_1O_2$. Find the locus of $O$ when $D$ moves across $BC$.
2010 AMC 10, 1
What is $ 100(100\minus{}3) \minus{} (100 \cdot 100 \minus{} 3)$?
$ \textbf{(A)}\ \minus{}20,000 \qquad \textbf{(B)}\ \minus{}10,000 \qquad \textbf{(C)}\ \minus{}297 \qquad \textbf{(D)}\ \minus{}6 \qquad \textbf{(E)}\ 0$
1984 IMO Longlists, 43
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
2007 Harvard-MIT Mathematics Tournament, 9
$\triangle ABC$ is right angled at $A$. $D$ is a point on $AB$ such that $CD=1$. $AE$ is the altitude from $A$ to $BC$. If $BD=BE=1$, what is the length of $AD$?
2011 AIME Problems, 9
Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p+q$.
2019 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.