Found problems: 700
Find the number of ordered pairs of integers $(m,n)$ such that $0 \le m,n \le 2023$ and $$m^2 \equiv \sum_{d \mid 2023} n^d \pmod{2024}.$$
$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow:
$L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$
[i]a)[/i] For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points)
[i]b)[/i] Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant.
Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix}
a & & &x \in A \\
b& & &x \in B \\
c& & & x \in C
\end{matrix}\right.$ . (7 points)
[i]c)[/i] Prove that $a+b+c = -3$. (4 points)
[i]d)[/i] Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points)
[i]e)[/i] Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points)
(${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
Determine all triples $(x,y,z)$ of positive integers such that
\[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
Find the remainder of
$$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$
when divided by $101$.
Prove that $\sqrt 2$ is irrational.
Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$
Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:
[list]
[*] $f(1,1)=0$.
[*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
[*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$.
[/list]
Prove that
$$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
Call an odd prime $p$ [i]adjective[/i] if there exists an infinite sequence $a_0,a_1,a_2,\ldots$ of positive integers such that \[a_0\equiv1+\frac{1}{a_1}\equiv1+\frac{1}{1+\frac{1}{a_2}}\equiv1+\frac{1}{1+\frac{1}{1+\frac{1}{a_3}}}\equiv\ldots\pmod p.\] What is the sum of the first three odd primes that are [i]not[/i] adjective?
Note: For two common fractions $\frac{a}{b}$ and $\frac{c}{d}$, we say that $\frac{a}{b}\equiv\frac{c}{d}\pmod p$ if $p$ divides $ad-bc$ and $p$ does not divide $bd$.
[i]2019 CCA Math Bonanza Individual Round #14[/i]
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.
Call a lattice point [i]visible[/i] if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]
Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals
$\textbf{(A) } 0\qquad \textbf{(B) }1 \qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } \text{not uniquely determined}$
Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.
For an integer $k$ let $T_k$ denote the number of $k$-tuples of integers $(x_1,x_2,...x_k)$ with $0\le x_i < 73$ for each $i$, such that $73|x_1^2+x_2^2+...+x_k^2-1$. Compute the remainder when $T_1+T_2+...+T_{2017}$ is divided by $2017$.
[i]Proposed by Vincent Huang
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3,\ldots$ such that $a_n \in \{1, 2,\ldots, p-1\}$ and $a_na_{n+2}\equiv1+a_{n+1}$ (mod $p$) for all $n\geq 1.$ Prove that $f(p)$ is congruent to 0 or 2 (mod 5).
At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it:
[list]
[*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$.
[*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$.
[/list]
What was the prime $p$?
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]