Practice (112)

2841

Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.

3070

If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?

3191

Prove that $7\mid 8^n-1$ for $n\ge 1$.

3192

Show that $5\mid 4^{2n}-1$ for $n\ge 1$.

3194

Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.

3743

An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?

3841

Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.

3971

Does there exist a polynomial $P(x)$ such that $P(1)=2015$ and $P(2015)=2016$?

4160

Let $n$ be any positive integer, show that $$(5n+1)(5n+2)(5n+3)(5n+4)\equiv -1 \pmod{25}$$

4172

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

4175

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.

4176

How many positive integers not exceeding $100$ are there such that the value of $(3^x-x^2)$ is a multiple of $5$?

4177

Let $\mathbb{S}$ be the set of integers between $1$ and $2^{40}$ that contain two $1$s when written in base $2$. What is the probability that a random integer from $\mathbb{S}$ is divisible by $9$?

4178

Find the multiplicative order of $3$ modulo $17$.

4179

Find the multiplicative order of $5$ modulo $19$.

4180

Show that if integer $a$ has multiplicative order of $hk$ modulo $n$, then $a^h$ has order of $k$ modulo $n$.

4181

Let $p$ be an odd prime, and integer $a$ has multiplicative order of $2k$ modulo $p$, then $a^k\equiv -1\pmod{p}$.

4182

Let $n$ be an odd integer greater than $1$, then $n$ is the multiplicative order of $2$ modulo $(2^n-1)$.

4186

(Thue's theorem) Let $p$ be a prime. Show that for any integer $a$ such that $p\not\mid a$, there exist positive integers $x$, $y$ not exceeding $\lfloor{\sqrt{p}}\rfloor$ satisfying $ax\equiv y\pmod{p}$ or $ax\equiv -y\pmod{p}$.

4187

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.

4188

Show that a prime $p > 2$ is a sum of two squares if and only if $p\equiv 1\pmod{4}$.

4190

(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.

4194

Find the smallest integer $N$ such that $\varphi(n) \ge 5$ holds for all integer $n \ge N$.

4195

Show that two positive integers $m$ and $n$ are co-prime if and only if $\varphi(mn)=\varphi(m)\varphi(n)$.

4196

Let $n > 4$ be a composite number. Show that $(n-1)!\equiv 0\pmod{n}$.

PolynomialAndEquation CombinatorialIdentity NumberTheoryBasic SquareNumber AlternativeBase EulerFermatTheorem MODBasic PigeonholePrinciple

AMC10/12 AIME Putnam

With Solutions

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