Found problems: 85335
Evaluate the sum of all three digits number which are not divisible by $13$ .
Given an integer $a>1$. Let $p_1 < p_2 <...< p_k$ be all prime divisors of $a$. For each positive integer $n$ we define:
$C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k}$
$A = a^2 + 1$
$T(n) = A^{C_0(n)} - 1$
$M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1)$
$A_n =\frac{T(n)}{M(n)}$
Prove that the sequence $A_1, A_2, ... $ satisfies the properties:
(i) Every number in the sequence is an integer greater than $1$ and has only prime divisors of the form $am + 1$.
(ii) Any two different numbers in the sequence are coprime.
A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks?
[i]Author: M.Ivanov, D.Rostovsky, V.Frank[/i]
Consider the equation
\[x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0\]
where $N$ is a given positive integer.
a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$),
b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.
Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.
There are $900$ students in an International School. There are $59$ international boys and $59$ international girls. The Students are partitioned into $30$ classrooms (each classrooms have equal number of student) and in each of the classrooms, the student will labelled number from $1$ to $30$. The Partition must satisfy at least one follow condition:
[list=i]
[*] Any Two international boys in same classroom can't have consecutive numbers.
[*] For every classroom, the student who is labelled $1$ must be a boy.
[/list]
Prove that there are $2$ classrooms, each of which has $2$ international boys with their labels difference equal.
Define a sequence $a_{m,n}$ where $a_{m,0}=1,$ and for all other $m,n$ (assuming $m \ge 1$): $$a_{m,n}=\begin{cases}
0 & n<0 \\
1 & n \equiv 0 \mod{m} \\
a_{m,n-1}+a_{m, n-m} & \text{else}
\end{cases}$$ If $\tfrac{a_{2025, (2025^2-1)}}{a_{2025, (2024^2-1)}} = \tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, then what is $a+b$?
If $p$, $q$, and $r$ are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of $(p+q+r)^2$.
[i]Proposed by David Altizio[/i]
For how many positive integers $n\le100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?
4031 lines are drawn on a plane, no two parallel or perpendicular, and no three lines meet at a point. Determine the maximum number of acute-angled triangles that may be formed.
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
The larger of two consecutive odd integers is three times the smaller. What is their sum?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$
Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$.
Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.
Let $C$ be a point lies outside the circle $(O)$ and $CS, CT$ are tangent lines of $(O)$. Take two points $A, B$ on $(O)$ with $M$ is the midpoint of the minor arc $AB$ such that $A, B, M$ differ from $S, T$. Suppose that $MS, MT$ cut line $AB$ at $E, F$. Take $X \in OS$ and $Y \in OT$ such that $EX, FY$ are perpendicular to $AB$. Prove that $X Y$ and $C M$ are perpendicular.
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$?
[asy]defaultpen(linewidth(1));
for ( int x = 0; x < 5; ++x )
{
draw((0,x)--(4,x));
draw((x,0)--(x,4));
}
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle);
fill((0,3)--(1,3)--(1,4)--(0,4)--cycle);
fill((2,3)--(4,3)--(4,4)--(2,4)--cycle);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle);
label("$A$", (0, 4), NW);
label("$B$", (4, 4), NE);
label("$C$", (4, 0), SE);
label("$D$", (0, 0), SW);[/asy]
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.
We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers.
For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$
[i]Proposed by Melih Üçer[/i]
The set of all points $(x,y)$ in the plane satisfying $x<y$ and $x^3-y^3>x^2-y^2$ has area $A$. What is the value of $A$?
[i]Proposed by Adam Bertelli[/i]
Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$
Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?