Found problems: 85335
Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying
\[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$
On the sides $AB, AC ,BC$ of the triangle $ABC$, the points $M, N, K$ are selected, respectively, such that $AM = AN$ and $BM = BK$. The circle circumscribed around the triangle $MNK$ intersects the segments $AB$ and $BC$ for the second time at points $P$ and $Q$, respectively. Lines $MN$ and $PQ$ intersect at point $T$. Prove that the line $CT$ bisects the segment $MP$.
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
Consider five spheres with radius $10$ cm . Four of these spheres are arranged on a horizontal table so that its centers form a $20$ cm square and the fifth sphere is placed on them so that it touches the other four. What is the distance between center of this fifth sphere and the table?
Suppose 2017 points in a plane are given such that no three points are collinear. Among the triangles formed by any three of these 2017 points, those triangles having the largest area are said to be [i]good[/i]. Prove that there cannot be more than 2017 good triangles.
Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$.
[i]Proposed by Prithwijit De[/i]
The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$?
$ \textbf{(A)}\ 1:3\qquad\textbf{(B)}\ 16:3\qquad\textbf{(C)}\ 20:3\qquad\textbf{(D)}\ 27:4\qquad\textbf{(E)}\ 12:1 $
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
$\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$
For real numbers $a_1, a_2, \ldots, a_{200}$, we consider the value $S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1$. In one operation, you can change the sign of any number (that is, change $a_i$ to $-a_i$), and then calculate the value of $S$ for the new numbers again. What is the smallest number of operations needed to always be able to make $S$ nonnegative?
[i]Proposed by Oleksii Masalitin[/i]
The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \begin{align*}2x+3y+3z&=8,\\3x+2y+3z&=808,\\3x+3y+2z&=80808.\end{align*} Their goal is not to find a solution $(x,y,z)$ to the system, but instead to compute the sum of the variables. Find the value of $x+y+z$.
The graphs of the following equations divide the $xy$ plane into some number of regions.
$4 + (x + 2)y =x^2$
$(x + 2)^2 + y^2 =16$
Find the area of the second smallest region.
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
In a village $1998$ persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to $3996$ feet. For this purpose, the field was divided into $1998$ equal parts. If each part had an integer area, find the length and breadth of the field.
The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.
Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]
Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, \quad a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$ \\Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$.
Prove that for positive real numbers $ a$, $ b$, $ c$, $ d$, we have \[ \frac{1}{\frac{1}{a}\plus{}\frac{1}{b}}\plus{}\frac{1}{\frac{1}{c}\plus{}\frac{1}{d}}\le\frac{1}{\frac{1}{a\plus{}c}\plus{}\frac{1}{b\plus{}d}}\]
For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.
Let $P$ be a point outside the circle $C$. Find two points $Q$ and $R$ on the circle, such that $P,Q$ and $R$ are collinear and $Q$ is the midpopint of the segmenet $PR$. (Discuss the number of solutions).
Devah draws a row of 1000 equally spaced dots on a sheet of paper. She goes through the dots from left to right, one by one, checking if the midpoint between the current dot and some remaining dot to its left is also a remaining dot. If so, she erases the current dot. How many dots does Devah end up erasing?
Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?
If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$ then $x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} = $
$ \textbf{(A)}\ 5.05\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 51.005\qquad\textbf{(D)}\ 61.25\qquad\textbf{(E)}\ 400 $
Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties:
[list]
[*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points,
[*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.