Found problems: 85335
Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.
Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes 8 equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make 9 equal rows with the remaining pebbles. After Kramer eats another, he finds he can make 10 equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.
An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.
Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x).$ There are relatively prime positive integers so that $x = -\tfrac{m}{n}$. Find $m + n.$
Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.
(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?
Each of a pair of opposite faces of a unit cube is marked by a dot. Each of another pair of opposite faces is marked by two dots. Each of the remaining two faces is marked by three dots. Eight such cubes are used to construct a $2\times 2 \times 2$ cube. Count the total number of dots on each of its six faces. Can we obtain six consecutive numbers?
(A Shapovalov)
Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information:
$i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$
$ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen:
a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows:
$a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$
b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$.
The sequence $C_0$ that appears in the screen is the following:
$a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$
Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.
Prove that the product of five consecutive positive integers cannot be a perfect square.
(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute .
(b) Same problem for a convex polyhedron with $n$ vertices.
(V. Boltyanskiy, Moscow)
Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now?
$\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
Let \[f(x)=\dfrac{1}{1-\dfrac{1}{1-x}}\,.\] Compute $f^{2016}(2016)$, where $f$ is composed upon itself $2016$ times.
In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then
$\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$
$\textbf{(D)}~\text{None of the above}$
What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros?
$\textbf{(A) }3\hspace{14em}\textbf{(B) }4\hspace{14em}\textbf{(C) }5$
$\textbf{(D) }6\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }8$
$\textbf{(G) }9\hspace{14em}\textbf{(H) }10\hspace{13.3em}\textbf{(I) }11$
$\textbf{(J) }12\hspace{13.8em}\textbf{(K) }13\hspace{13.3em}\textbf{(L) }14$
$\textbf{(M) }2007$
There are $9$ point in the Cartezian plane with coordinates
$(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$
Some points are coloured in red and the others in blue. Prove that for any colouring of the points we can always find a right isosceles triangle whose vertexes have the same colour.
Square $ABCD$ with side-length $n$ is divided into $n^2$ small (fundamental) squares by drawing lines parallel to its sides (the case $n=5$ is presented on the diagram).The squares' vertices that lie inside (or on the boundary) of the triangle $ABD$ are connected with each other with arcs.Starting from $A$,we move only upwards or to the right.Each movement takes place on the segments that are defined by the fundamental squares and the arcs of the circles.How many possible roots are there in order to reach $C$;
Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Powell Zhang[/i]
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
Inside a convex quadrilateral $ABCD$ with $BC>AD$, a point $T$ is chosen. $S$ lies on the segment $AT$ such that $DT = BC, \angle TSD = 90^\circ$.
Prove that if $\angle DTA + \angle TAB + \angle ABC = 180^\circ$, then $AB + ST \geqslant CD + AS$.
[i]Proposed by Alexander Tereshin - Russia[/i]
Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that
\[n+f(m)\mid f(n)+nf(m)\]
for all $m,n\in \mathbb{N}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $a, b, c$ be positive real numbers such that $abc = \dfrac {1} {8}$. Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}$$
When the equality holds?
When Tim was Jim’s age, Kim was twice as old as Jim. When Kim was Tim’s age, Jim was 30. When Jim becomes Kim’s age, Tim will be 88. When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?