Found problems: 1048
Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number.
[b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties
\[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \]
Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
Let $ p_1,\dots,p_k$ be prime numbers, and let $ S$ be the set of those integers whose all prime divisors are among $ p_1,\dots,p_k$. For a finite subset $ A$ of the integers let us denote by $ \mathcal G(A)$ the graph whose vertices are the elements of $ A$, and the edges are those pairs $ a,b\in A$ for which $ a \minus{} b\in S$. Does there exist for all $ m\geq 3$ an $ m$-element subset $ A$ of the integers such that
(i) $ \mathcal G(A)$ is complete?
(ii) $ \mathcal G(A)$ is connected, but all vertices have degree at most 2?
Find all the triples of positive integers $(a,b,c)$ for which the number
\[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\]
is an integer and $a+b+c$ is a prime.
Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$ and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$
There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression.
Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.
Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.
Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
Let $p \equiv 3 \,(\textrm{mod}\, 4)$ be a prime and $\theta$ some angle such that $\tan(\theta)$ is rational. Prove that $\tan((p+1)\theta)$ is a rational number with numerator divisible by $p$, that is, $\tan((p+1)\theta) = \frac{u}{v}$ with $u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1$ and $u \equiv 0 \,(\textrm{mod}\,p) $.
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?
[asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle);
draw(card, linewidth(1));
draw(shift(2.5,0)*card, linewidth(1));
draw(shift(5,0)*card, linewidth(1));
label("$44$", (1,1.5));
label("$59$", shift(2.5,0)*(1,1.5));
label("$38$", shift(5,0)*(1,1.5));[/asy]
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.
Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$
a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number.
b) $x^{2^n}+1,$ with $n$ is a positive integer.
In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime.
$$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$
[asy]
real HRT3 = sqrt(3) / 2;
void drawCircle(real x, real y, real r) {
path p = circle((x,y), r);
draw(p);
fill(p, white);
}
void drawCell(int gx, int gy) {
real x = 0.5 * gx;
real y = HRT3 * gy;
drawCircle(x, y, 0.35);
}
void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) {
real x1 = 0.5 * gx1;
real y1 = HRT3 * gy1;
real x2 = 0.5 * gx2;
real y2 = HRT3 * gy2;
if (doubled) {
real dx = x2 - x1;
real dy = y2 - y1;
real ox = -0.035 * dy / sqrt(dx * dx + dy * dy);
real oy = 0.035 * dx / sqrt(dx * dx + dy * dy);
draw((x1+ox,y1+oy)--(x2+ox,y2+oy));
draw((x1-ox,y1-oy)--(x2-ox,y2-oy));
} else {
draw((x1,y1)--(x2,y2));
}
}
drawEdge(2, 0, 4, 0, true);
drawEdge(2, 0, 1, 1, true);
drawEdge(2, 0, 3, 1, true);
drawEdge(4, 0, 3, 1, false);
drawEdge(4, 0, 5, 1, false);
drawEdge(1, 1, 0, 2, false);
drawEdge(1, 1, 2, 2, false);
drawEdge(1, 1, 3, 1, false);
drawEdge(3, 1, 2, 2, true);
drawEdge(3, 1, 4, 2, true);
drawEdge(3, 1, 5, 1, false);
drawEdge(5, 1, 4, 2, true);
drawEdge(5, 1, 6, 2, false);
drawEdge(0, 2, 1, 3, false);
drawEdge(0, 2, 2, 2, false);
drawEdge(2, 2, 1, 3, false);
drawEdge(2, 2, 3, 3, true);
drawEdge(2, 2, 4, 2, false);
drawEdge(4, 2, 3, 3, false);
drawEdge(4, 2, 5, 3, false);
drawEdge(4, 2, 6, 2, false);
drawEdge(6, 2, 5, 3, true);
drawEdge(1, 3, 3, 3, true);
drawEdge(3, 3, 5, 3, false);
drawCell(2, 0);
drawCell(4, 0);
drawCell(1, 1);
drawCell(3, 1);
drawCell(5, 1);
drawCell(0, 2);
drawCell(2, 2);
drawCell(4, 2);
drawCell(6, 2);
drawCell(1, 3);
drawCell(3, 3);
drawCell(5, 3);
[/asy]
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Let $p$ be a prime number and $a$ be an integer.
Prove that if $2^p +3^p = a^n$ for some integer $n$, then $n = 1$.
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct.
Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.
(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers.
(b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.