Found problems: 85335
Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$. Find $m+n$
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.
Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.
For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$
Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .
Rhombus $ABCD$ has side length $ 1$. The size of $\angle A$ (in degrees) is randomly selected from all real numbers between $0$ and $90$. Find the expected value of the area of $ABCD$.
Let $ABC$ be a triangle with $AB \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$
A permutation of the integers \(2020, 2021,...,2118, 2119\) is a list \(a_1,a_2,a_3,...,a_{100}\) where each one of the numbers appears exactly once. For each permutation we define the partial sums.
$s_1=a_1$
$s_2=a_1+a_2$
$s_3=a_1+a_2+a_3$
$...$
$s_{100}=a_1+a_2+...+a_{100}$
How many of these permutations satisfy that none of the numbers \(s_1,...,s_{100}\) is divisible by $3$?
The base of the pyramid is a triangle $ABC$, in which $\angle ACB= 30^o$, and the length of the median from the vertex $B$ is twice less than the side $AC$ and is equal to $\alpha$ . All side edges of the pyramid are inclined to the plane of the base at an angle $a$. Determine the cross-sectional area of the pyramid with a plane passing through the vertex $B$ parallel to the edge $AD$ and inclined to the plane of the base at an angle of $\beta$,
Prove that the equation
$$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
Set $T=\{0,1,2,3,4,5,6\},M=\left\{\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4}|a_i\in T,i=1,2,3,4\right\}$. Put all elements in $M$ in order: from small to large, then the 2005th number is
$\text{(A)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{3}{7^4}$
$\text{(B)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{2}{7^4}$
$\text{(C)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{4}{7^4}$
$\text{(D)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{3}{7^4}$
Find all polynomials $P(x)$ with real coefficients satisfying the equation \[(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)\] for all real numbers $x$.
Let $\Gamma$ be a circle of radius $1$ centered at $O$. A circle $\Omega$ is said to be \emph{friendly} if there exist distinct circles $\omega_1$, $\omega_2$, $\ldots$, $\omega_{2020}$, such that for all $1\le i\le2020$, $\omega_i$ is tangent to $\Gamma$, $\Omega$, and $\omega_{i+1}$. (Here, $\omega_{2021} = \omega_1$.) For each point $P$ in the plane, let $f(P)$ denote the sum of the areas of all friendly circles centered at $P$. If $A$ and $B$ are points such that $OA=\frac12$ and $OB=\frac13$, determine $f(A)-f(B)$.
[i]Proposed by Michael Ren.[/i]
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac13 \qquad
\textbf{(C)}\ \frac12 \qquad
\textbf{(D)}\ \frac23 \qquad
\textbf{(E)}\ \frac34$
In a school there are 2008 students. Students are members of certain committees. A committee has at most 1004 members and every two students join a common committee.
(a) Determine the smallest possible number of committees in the school.
(b) If it is further required that the union of any two committees consists of at most 1800 students, will your answer in (a) still hold?
An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:
(a) there is no silver matrix for $ n \equal{} 1997$;
(b) silver matrices exist for infinitely many values of $ n$.
Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that
\[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ :
$[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$.
Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that
(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.
(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.
( V . V . Proizvolov , Moscow)
Three given circles have the same radius and pass through a common point $ P$. Their other points of pairwise intersections are $ A$, $ B$, $ C$. We define triangle $ A'B'C'$, each of whose sides is tangent to two of the three circles. The three circles are contained in $ \triangle A'B'C'$. Prove that the area of $ \triangle A'B'C'$ is at least nine times the area of $ \triangle ABC$
Region $ABCDEFGHIJ$ consists of $13$ equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28);
draw(P--Q--R--S--cycle);
picture p = new picture;
draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5));
draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3));
transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5);
add(t*p);
label("$P$",P,SW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,SE); label("$A$",t*(0,-3),W); label("$B$",B,N); label("$C$",t*(3,0),plain.ENE); label("$D$",t*(3,-2),NE); label("$E$",t*(5,-2),plain.E); label("$F$",t*(5,-3),plain.SW); label("$G$",t*(3,-3),(0.81,-1.3)); label("$H$",t*(3,-5),plain.S); label("$I$",t*(2,-5),NW); label("$J$",t*(2,-3),SW);[/asy]
Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$