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Stage 5 · Number theory

10 problems · AIME late · mathsolympiadprep.com

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1. Veronica observes that $81 \cdot 3=243$ and $81 \cdot 4=324$ and wonders how many numbers $m$ there are with $10 \leq m \leq 99$ such that $3 m=A B C$ and $4 m=C A B$, with $A, B$ and $C$ decimal digits (the cases in which one or more of the digits $A, B, C$ are equal to zero are also considered valid).

   1. A1
   2. B2
   3. C3
   4. D4
   5. E6

   Progetto Olimpiadi della Matematica - GARA di FEBBRAIO

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2. For positive integers $n$, $m$, $k$, let us write $n \equiv m \pmod k$ if $n - m$ is divisible by $k$. Let $A$ be the sum of all positive integers $a$ less than or equal to $2011$ for which $a \equiv 1 \pmod 3$, and $B$ be the sum of all positive integers $b$ less than or equal to $2011$ for which $b \equiv 2 \pmod 3$. Find the value of $A - B$.

   Japan Mathematical Olympiad

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3. What is the last digit of $1^{1}+2^{2}+3^{3}+\cdots+100^{100}$?

   HMMT February

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4. For any positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\frac{\tau\left(n^{2}\right)}{\tau(n)}=3$, compute $\frac{\tau\left(n^{7}\right)}{\tau(n)}$.

   HMMT February

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5. A prime number $p$ is twin if at least one of $p+2$ or $p-2$ is prime and sexy if at least one of $p+6$ and $p-6$ is prime.
   How many sexy twin primes (i.e. primes that are both twin and sexy) are there less than $10^{9}$? Express your answer as a positive integer $N$ in decimal notation; for example, 521495223. If your answer is in this form, your score for this problem will be $\max \left\{0,25-\left\lfloor\frac{1}{10000}|A-N|\right\rfloor\right\}$, where $A$ is the actual answer to this problem. Otherwise, your score will be zero.

   HMMT February

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6. Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a} \underline{b} \underline{c}, if \underline{a} \underline{b} \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b} \underline{c} \underline{a}$ is also a multiple of $x$.

   HMMT February

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7. Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \in\{0,1, \ldots, p-1\}$ and $p \text{ divides } y^{2}-x^{3}-4x$.

   HMMT February

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8. Luca writes on a blackboard all the possible sequences consisting of 2017 distinct positive integers whose sum is $2016 \cdot 2017 \cdot 2018$. Having done this, he replaces each of these sequences with the greatest common divisor of its elements. When this lengthy operation is finished, what is the maximum among the numbers written on the blackboard?

   1. A2
   2. B$2 \cdot 2016$
   3. C$2 \cdot 2017$
   4. D$2 \cdot 2018$
   5. E$2016 \cdot 2018$

   Gara di Febbraio

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9. Let $m$ be a positive integer. Let $d(n)$ denote the number of divisors of $n$, and define the function $F(x)=\sum_{n=1}^{105^{m}} \frac{d(n)}{n^{x}}$. Define the numbers $a(n)$ to be the positive integers for which $F(x)^{2}=\sum_{n=1}^{105^{2 m}} \frac{a(n)}{n^{x}}$ for all real $x$. Express $a\left(105^{m}\right)$ in terms of $m$.

   HMMT February

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10. Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

    Junior Balkan MO Shortlist

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Answer key — Stage 5 · Number theory

Worked solutions for every problem are on the site, one page per problem.

1. $C$ open
2. $1341$ open
3. $0$ open
4. $29$ open
5. $1462105$ open
6. $64$ open
7. $p=2 \text{ and } p \equiv 3(\bmod 4)$ open
8. $B$ open
9. $\left(\frac{m^{3}+6 m^{2}+11 m+6}{6}\right)^{3} \text{ OR } \binom{m+3}{3}^{3}$ open
10. $83$ open

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