Found problems: 700
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.
Prove that the equation $x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$ has no solution in the set of the positive integers.
Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.
Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying:
$ p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3)$.
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$.
(b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.
Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.
Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that
\begin{align*}
x^2+y^2+z^2 &\equiv 0 \pmod n;\\
xyz &\equiv k \pmod n.
\end{align*}
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[b]B1[/b] What is the sum of the first $5$ positive integers?
[b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$?
[b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number?
[b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen?
[b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$?
[b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle?
[b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus?
[b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win?
[b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece?
[b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$?
[b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime?
Note: $1$ is not prime.
[b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement?
[b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares?
[b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions?
[b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card?
[b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$?
[b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$?
[b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$?
[b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$.
[b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find all triples of non-negative integers \((a, b, c)\) such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]
Prove that there are no perfect squares in the array below:
\[\begin{array}{cccc}11&111&1111&...\\22&222&2222&...\\33&333&3333&...\\44&444&4444&...\\55&555&5555&... \\66&666&6666&...\\77&777&7777&...\\88&888&8888&...\\99&999&9999&...\end{array}\]
Let $a$, $b$, $c$ be positive integers with $a \le 10$. Suppose the parabola $y = ax^2 + bx + c$ meets the $x$-axis at two distinct points $A$ and $B$. Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$.
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
[/hide]
$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that
$$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$
[i]Proposed by Ali Partofard[/i]
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation
$$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$
Let $p$ be an odd prime. An integer $x$ is called a [i]quadratic non-residue[/i] if $p$ does not divide $x-t^2$ for any integer $t$.
Denote by $A$ the set of all integers $a$ such that $1\le a<p$, and both $a$ and $4-a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$.
[i]Proposed by Richard Stong and Toni Bluher[/i]
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$
(a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$
(b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$
[I]United Kingdom[/i]
Determine all $2\times2$ integer matrices $A$ having the following properties:
$1.$ the entries of $A$ are (positive) prime numbers,
$2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$
Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares.
[i](Proposed by Orestis Lignos, Greece)[/i]