Found problems: 85335
Find all real solutions of $\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}$
It is known that every student of the class for Sunday once visited the rink, and every boy met there with every girl. Prove that there was a point in time when all the boys, or all the girls of the class were simultaneously on the rink.
For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that
\[p_1^2+p_2^2+\ldots +p_k^2=2010? \]
Compute the number of triangles of different sizes which contain the gray triangle in the figure below.
[asy]
size(5cm);
real n = 4;
for (int i = 0; i < n; ++i) {
draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
draw((i,0)--((n+i)/2,(n-i)*0.866));
}
filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray);
[/asy]
[i]Proposed by Nathan Xiong[/i]
a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?
Find all $(x,y)$ real numbers pairs such that,
$$x^7+y^7=x^4+y^4$$
Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent).
Find the minimum possible value of the radii of the semi-circles.
At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers
\begin{align*}
a'&=a+4b+16c+64d\\
b'&=b+4c+16d+64a\\
c'&=c+4d+16a+64b\\
d'&=d+4a+16b+64c
\end{align*}
and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$.
For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?
In a circle with center $O$, chords $AB$ and $AC$ are drawn, both equal to the radius. Points $A_1$, $B_1$ and $C_1$ are projections of points $A, B$ and $C$, respectively, onto an arbitrary diameter $XY$. Prove that one of the segments $XB_1$, $OA_1$ and $C_1Y$ is equal to the sum of the other two.
(A. Shklover)
A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression:
(b-a)(b^3+3a^3)
Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result?
In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.
A positive integer $m$ is called [i]growing[/i] if its digits, read from left to right, are non-increasing. Prove that for each natural number $n$ there exists a growing number $m$ with $n$ digits such that the sum of its digits is a perfect square.
Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property
\[ d( f(A),f(B)) \geq d(A,B) \] for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.
Prove that:
i)
$$\sum_{k=1}^{n-1}(1+\ln k)\le n^2-n+1$$
ii)
$$\sum_{k=1}^{n-1}\sqrt{\ln k}\le\frac{n^2-n+1}2$$
[i]Proposed by Laurențiu Modan[/i]
Let $ n$, $ k$, $ p$ be positive integers with $ 2 \le k \le \frac {n}{p \plus{} 1}$. Let $ n$ distinct points on a circle be given. These points are colored blue and red so that exactly $ k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $ p$ red points. How many such colorings are there?
If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.
Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$
The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.
Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for
a) $N = 2011$
b) $N = 2012$ ?
Jim starts with a positive integer $ n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $ n=55$, then his sequence contains $ 5$ numbers:
\begin{align*}
&55\\
55-7^2=&\ 6\\
6-2^2=&\ 2\\
2-1^2=&\ 1\\
1-1^2=&\ 0
\end{align*}Let $ N$ be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of $ N$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 9$