id int32 | title string | problem string | question_latex string | question_html string | numerical_answer string | pub_date string | solved_by string | diff_rate string | difficulty string |
|---|---|---|---|---|---|---|---|---|---|
733 | Ascending Subsequences | Let $a_i$ be the sequence defined by $a_i=153^i \bmod 10\,000\,019$ for $i \ge 1$.
The first terms of $a_i$ are:
$153, 23409, 3581577, 7980255, 976697, 9434375, \dots$
Consider the subsequences consisting of $4$ terms in ascending order. For the part of the sequence shown above, these are:
$153, 23409, 3581577, 79802... | Let $a_i$ be the sequence defined by $a_i=153^i \bmod 10\,000\,019$ for $i \ge 1$.
The first terms of $a_i$ are:
$153, 23409, 3581577, 7980255, 976697, 9434375, \dots$
Consider the subsequences consisting of $4$ terms in ascending order. For the part of the sequence shown above, these are:
$153, 23409, 3581577, 79802... | <p>
Let $a_i$ be the sequence defined by $a_i=153^i \bmod 10\,000\,019$ for $i \ge 1$.<br/>
The first terms of $a_i$ are:
$153, 23409, 3581577, 7980255, 976697, 9434375, \dots$
</p>
<p>
Consider the subsequences consisting of $4$ terms in ascending order. For the part of the sequence shown above, these are:<br/>
$153, ... | 574368578 | Saturday, 7th November 2020, 04:00 pm | 509 | 25% | easy |
484 | Arithmetic Derivative | The arithmetic derivative is defined by
$p^\prime = 1$ for any prime $p$
$(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule)
For example, $20^\prime = 24$.
Find $\sum \operatorname{\mathbf{gcd}}(k,k^\prime)$ for $1 \lt k \le 5 \times 10^{15}$.
Note: $\operatorname{\mathbf{gcd}}(x,y)$ denotes t... | The arithmetic derivative is defined by
$p^\prime = 1$ for any prime $p$
$(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule)
For example, $20^\prime = 24$.
Find $\sum \operatorname{\mathbf{gcd}}(k,k^\prime)$ for $1 \lt k \le 5 \times 10^{15}$.
Note: $\operatorname{\mathbf{gcd}}(x,y)$ denotes t... | <p>The <strong>arithmetic derivative</strong> is defined by</p>
<ul><li>$p^\prime = 1$ for any prime $p$</li>
<li>$(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule)</li>
</ul><p>For example, $20^\prime = 24$.</p>
<p>Find $\sum \operatorname{\mathbf{gcd}}(k,k^\prime)$ for $1 \lt k \le 5 \times... | 8907904768686152599 | Saturday, 11th October 2014, 01:00 pm | 441 | 100% | hard |
6 | Sum Square Difference | The sum of the squares of the first ten natural numbers is,
$$1^2 + 2^2 + ... + 10^2 = 385.$$
The square of the sum of the first ten natural numbers is,
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025.$$
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025 - 385 =... | The sum of the squares of the first ten natural numbers is,
$$1^2 + 2^2 + ... + 10^2 = 385.$$
The square of the sum of the first ten natural numbers is,
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025.$$
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025 - 385 =... | <p>The sum of the squares of the first ten natural numbers is,</p>
$$1^2 + 2^2 + ... + 10^2 = 385.$$
<p>The square of the sum of the first ten natural numbers is,</p>
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025.$$
<p>Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum... | 25164150 | Friday, 14th December 2001, 06:00 pm | 525978 | 5% | easy |
598 | Split Divisibilities | Consider the number $48$.
There are five pairs of integers $a$ and $b$ ($a \leq b$) such that $a \times b=48$: $(1,48)$, $(2,24)$, $(3,16)$, $(4,12)$ and $(6,8)$.
It can be seen that both $6$ and $8$ have $4$ divisors.
So of those five pairs one consists of two integers with the same number of divisors.
In general:
Le... | Consider the number $48$.
There are five pairs of integers $a$ and $b$ ($a \leq b$) such that $a \times b=48$: $(1,48)$, $(2,24)$, $(3,16)$, $(4,12)$ and $(6,8)$.
It can be seen that both $6$ and $8$ have $4$ divisors.
So of those five pairs one consists of two integers with the same number of divisors.
In general:
Le... | <p>
Consider the number $48$.<br/>
There are five pairs of integers $a$ and $b$ ($a \leq b$) such that $a \times b=48$: $(1,48)$, $(2,24)$, $(3,16)$, $(4,12)$ and $(6,8)$.<br/>
It can be seen that both $6$ and $8$ have $4$ divisors.<br/>
So of those five pairs one consists of two integers with the same number of diviso... | 543194779059 | Sunday, 9th April 2017, 10:00 am | 538 | 40% | medium |
700 | Eulercoin | Leonhard Euler was born on 15 April 1707.
Consider the sequence 1504170715041707n mod 4503599627370517.
An element of this sequence is defined to be an Eulercoin if it is strictly smaller than all previously found Eulercoins.
For example, the first term is 1504170715041707 which is the first Eulercoin. The second term... | Leonhard Euler was born on 15 April 1707.
Consider the sequence 1504170715041707n mod 4503599627370517.
An element of this sequence is defined to be an Eulercoin if it is strictly smaller than all previously found Eulercoins.
For example, the first term is 1504170715041707 which is the first Eulercoin. The second term... | <p>Leonhard Euler was born on 15 April 1707.</p>
<p>Consider the sequence 1504170715041707<var>n</var> mod 4503599627370517.</p>
<p>An element of this sequence is defined to be an Eulercoin if it is strictly smaller than all previously found Eulercoins.</p>
<p>For example, the first term is 1504170715041707 which is th... | 1517926517777556 | Saturday, 1st February 2020, 01:00 pm | 3887 | 5% | easy |
881 | Divisor Graph Width | For a positive integer $n$ create a graph using its divisors as vertices. An edge is drawn between two vertices $a \lt b$ if their quotient $b/a$ is prime. The graph can be arranged into levels where vertex $n$ is at level $0$ and vertices that are a distance $k$ from $n$ are on level $k$. Define $g(n)$ to be the maxim... | For a positive integer $n$ create a graph using its divisors as vertices. An edge is drawn between two vertices $a \lt b$ if their quotient $b/a$ is prime. The graph can be arranged into levels where vertex $n$ is at level $0$ and vertices that are a distance $k$ from $n$ are on level $k$. Define $g(n)$ to be the maxim... | <p>
For a positive integer $n$ create a graph using its divisors as vertices. An edge is drawn between two vertices $a \lt b$ if their quotient $b/a$ is prime. The graph can be arranged into levels where vertex $n$ is at level $0$ and vertices that are a distance $k$ from $n$ are on level $k$. Define $g(n)$ to be the m... | 205702861096933200 | Saturday, 9th March 2024, 10:00 pm | 502 | 20% | easy |
443 | GCD Sequence | Let $g(n)$ be a sequence defined as follows:
$g(4) = 13$,
$g(n) = g(n-1) + \gcd(n, g(n-1))$ for $n \gt 4$.
The first few values are:
$n$4567891011121314151617181920...
$g(n)$1314161718272829303132333451545560...
You are given that $g(1\,000) = 2524$ and $g(1\,000\,000) = 2624152$.
Find $g(10^{15})$. | Let $g(n)$ be a sequence defined as follows:
$g(4) = 13$,
$g(n) = g(n-1) + \gcd(n, g(n-1))$ for $n \gt 4$.
The first few values are:
$n$4567891011121314151617181920...
$g(n)$1314161718272829303132333451545560...
You are given that $g(1\,000) = 2524$ and $g(1\,000\,000) = 2624152$.
Find $g(10^{15})$. | <p>Let $g(n)$ be a sequence defined as follows:<br/>
$g(4) = 13$,<br/>
$g(n) = g(n-1) + \gcd(n, g(n-1))$ for $n \gt 4$.</p>
<p>The first few values are:</p>
<div align="center">
<table align="center" border="0" cellpadding="5" cellspacing="1"><tr><td>$n$</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><... | 2744233049300770 | Saturday, 2nd November 2013, 04:00 pm | 1282 | 30% | easy |
818 | SET | The SET® card game is played with a pack of $81$ distinct cards. Each card has four features (Shape, Color, Number, Shading). Each feature has three different variants (e.g. Color can be red, purple, green).
A SET consists of three different cards such that each feature is either the same on each card or different on ... | The SET® card game is played with a pack of $81$ distinct cards. Each card has four features (Shape, Color, Number, Shading). Each feature has three different variants (e.g. Color can be red, purple, green).
A SET consists of three different cards such that each feature is either the same on each card or different on ... | <p>
The SET® card game is played with a pack of $81$ distinct cards. Each card has four features (Shape, Color, Number, Shading). Each feature has three different variants (e.g. Color can be red, purple, green).</p>
<p>
A <i>SET</i> consists of three different cards such that each feature is either the same on each car... | 11871909492066000 | Sunday, 27th November 2022, 10:00 am | 147 | 85% | hard |
386 | Maximum Length of an Antichain | Let $n$ be an integer and $S(n)$ be the set of factors of $n$.
A subset $A$ of $S(n)$ is called an antichain of $S(n)$ if $A$ contains only one element or if none of the elements of $A$ divides any of the other elements of $A$.
For example: $S(30) = \{1, 2, 3, 5, 6, 10, 15, 30\}$.
$\{2, 5, 6\}$ is not an antichain of $... | Let $n$ be an integer and $S(n)$ be the set of factors of $n$.
A subset $A$ of $S(n)$ is called an antichain of $S(n)$ if $A$ contains only one element or if none of the elements of $A$ divides any of the other elements of $A$.
For example: $S(30) = \{1, 2, 3, 5, 6, 10, 15, 30\}$.
$\{2, 5, 6\}$ is not an antichain of $... | <p>Let $n$ be an integer and $S(n)$ be the set of factors of $n$.</p>
<p>A subset $A$ of $S(n)$ is called an <strong>antichain</strong> of $S(n)$ if $A$ contains only one element or if none of the elements of $A$ divides any of the other elements of $A$.</p>
<p>For example: $S(30) = \{1, 2, 3, 5, 6, 10, 15, 30\}$.
<br/... | 528755790 | Sunday, 27th May 2012, 08:00 am | 833 | 40% | medium |
333 | Special Partitions | All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as $2^i \times 3^j$, where $i,j \ge 0$.
Let's consider only such partitions where none of the terms can divide any of the other terms.
For example, the partition of $17 = 2 + 6 + 9 = (2^1 \times 3^0 + 2^1 \... | All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as $2^i \times 3^j$, where $i,j \ge 0$.
Let's consider only such partitions where none of the terms can divide any of the other terms.
For example, the partition of $17 = 2 + 6 + 9 = (2^1 \times 3^0 + 2^1 \... | <p>All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as $2^i \times 3^j$, where $i,j \ge 0$.</p>
<p>Let's consider only such partitions where none of the terms can divide any of the other terms.
<br/>For example, the partition of $17 = 2 + 6 + 9 = (2^1 \ti... | 3053105 | Saturday, 16th April 2011, 01:00 pm | 1374 | 35% | medium |
791 | Average and Variance | Denote the average of $k$ numbers $x_1, ..., x_k$ by $\bar{x} = \frac{1}{k} \sum_i x_i$. Their variance is defined as $\frac{1}{k} \sum_i \left( x_i - \bar{x} \right) ^ 2$.
Let $S(n)$ be the sum of all quadruples of integers $(a,b,c,d)$ satisfying $1 \leq a \leq b \leq c \leq d \leq n$ such that their average is exactl... | Denote the average of $k$ numbers $x_1, ..., x_k$ by $\bar{x} = \frac{1}{k} \sum_i x_i$. Their variance is defined as $\frac{1}{k} \sum_i \left( x_i - \bar{x} \right) ^ 2$.
Let $S(n)$ be the sum of all quadruples of integers $(a,b,c,d)$ satisfying $1 \leq a \leq b \leq c \leq d \leq n$ such that their average is exactl... | <p>Denote the average of $k$ numbers $x_1, ..., x_k$ by $\bar{x} = \frac{1}{k} \sum_i x_i$. Their variance is defined as $\frac{1}{k} \sum_i \left( x_i - \bar{x} \right) ^ 2$.</p>
<p>Let $S(n)$ be the sum of all quadruples of integers $(a,b,c,d)$ satisfying $1 \leq a \leq b \leq c \leq d \leq n$ such that their average... | 404890862 | Sunday, 27th March 2022, 02:00 am | 168 | 60% | hard |
360 | Scary Sphere | Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in three dimensional space, the Manhattan distance between those points is defined as$|x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|$.
Let $C(r)$ be a sphere with radius $r$ and center in the origin $O(0,0,0)$.
Let $I(r)$ be the set of all points with integer coordina... | Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in three dimensional space, the Manhattan distance between those points is defined as$|x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|$.
Let $C(r)$ be a sphere with radius $r$ and center in the origin $O(0,0,0)$.
Let $I(r)$ be the set of all points with integer coordina... | <p>
Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in three dimensional space, the <strong>Manhattan distance</strong> between those points is defined as<br/>$|x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|$.
</p>
<p>
Let $C(r)$ be a sphere with radius $r$ and center in the origin $O(0,0,0)$.<br/>
Let $I(r)$ be the s... | 878825614395267072 | Sunday, 27th November 2011, 01:00 am | 651 | 50% | medium |
591 | Best Approximations by Quadratic Integers | Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision $10^{-13}$:
$$4375636191520\sqrt{2}-6188084046055 < \pi < 721133315582\sqrt{2}-1019836515172 $$
We c... | Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision $10^{-13}$:
$$4375636191520\sqrt{2}-6188084046055 < \pi < 721133315582\sqrt{2}-1019836515172 $$
We c... | <p>Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by <b>quadratic integers</b> $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision $10^{-13}$:<br>
$$4375636191520\sqrt{2}-6188084046055 < \pi < 721133315582\sqrt{2}-10... | 526007984625966 | Saturday, 18th February 2017, 01:00 pm | 218 | 95% | hard |
505 | Bidirectional Recurrence | Let:
$\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text { } \rfloor \text { is the floor function} \\ x(2k+1)&=(2x(k)+3x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \\ y_n(k)&=\left\{{\beg... | Let:
$\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text { } \rfloor \text { is the floor function} \\ x(2k+1)&=(2x(k)+3x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \\ y_n(k)&=\left\{{\beg... | <p>Let:</p>
<p style="margin-left:32px;">$\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text { } \rfloor \text { is the floor function} \\ x(2k+1)&=(2x(k)+3x(\lfloor \frac k 2 \rfloor)) \text{ mod }... | 714591308667615832 | Sunday, 1st March 2015, 01:00 am | 232 | 90% | hard |
890 | Binary Partitions | Let $p(n)$ be the number of ways to write $n$ as the sum of powers of two, ignoring order.
For example, $p(7) = 6$, the partitions being
$$
\begin{align}
7 &= 1+1+1+1+1+1+1 \\
&=1+1+1+1+1+2 \\
&=1+1+1+2+2 \\
&=1+1+1+4 \\
&=1+2+2+2 \\
&=1+2+4
\end{align}
$$
You are also given $p(7^7) \equiv 144548435 \pmod {10^9+7}$.
Fi... | Let $p(n)$ be the number of ways to write $n$ as the sum of powers of two, ignoring order.
For example, $p(7) = 6$, the partitions being
$$
\begin{align}
7 &= 1+1+1+1+1+1+1 \\
&=1+1+1+1+1+2 \\
&=1+1+1+2+2 \\
&=1+1+1+4 \\
&=1+2+2+2 \\
&=1+2+4
\end{align}
$$
You are also given $p(7^7) \equiv 144548435 \pmod {10^9+7}$.
Fi... | <p>Let $p(n)$ be the number of ways to write $n$ as the sum of powers of two, ignoring order.</p>
<p>For example, $p(7) = 6$, the partitions being
$$
\begin{align}
7 &= 1+1+1+1+1+1+1 \\
&=1+1+1+1+1+2 \\
&=1+1+1+2+2 \\
&=1+1+1+4 \\
&=1+2+2+2 \\
&=1+2+4
\end{align}
$$
You are also given $p(7^7) \e... | 820442179 | Sunday, 12th May 2024, 02:00 am | 192 | 55% | medium |
376 | Nontransitive Sets of Dice | Consider the following set of dice with nonstandard pips:
Die $A$: $1$ $4$ $4$ $4$ $4$ $4$
Die $B$: $2$ $2$ $2$ $5$ $5$ $5$
Die $C$: $3$ $3$ $3$ $3$ $3$ $6$
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
If the first player picks die $A$ and the s... | Consider the following set of dice with nonstandard pips:
Die $A$: $1$ $4$ $4$ $4$ $4$ $4$
Die $B$: $2$ $2$ $2$ $5$ $5$ $5$
Die $C$: $3$ $3$ $3$ $3$ $3$ $6$
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
If the first player picks die $A$ and the s... | <p>
Consider the following set of dice with nonstandard pips:
</p>
<p>
Die $A$: $1$ $4$ $4$ $4$ $4$ $4$<br/>
Die $B$: $2$ $2$ $2$ $5$ $5$ $5$<br/>
Die $C$: $3$ $3$ $3$ $3$ $3$ $6$<br/></p>
<p>
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
</p>
<p>
If ... | 973059630185670 | Sunday, 18th March 2012, 01:00 am | 319 | 70% | hard |
115 | Counting Block Combinations II | NOTE: This is a more difficult version of Problem 114.
A row measuring $n$ units in length has red blocks with a minimum length of $m$ units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fill-count function, $F(m, n)$, represen... | NOTE: This is a more difficult version of Problem 114.
A row measuring $n$ units in length has red blocks with a minimum length of $m$ units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fill-count function, $F(m, n)$, represen... | <p class="note">NOTE: This is a more difficult version of <a href="problem=114">Problem 114</a>.</p>
<p>A row measuring $n$ units in length has red blocks with a minimum length of $m$ units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.... | 168 | Friday, 24th February 2006, 06:00 pm | 11053 | 35% | medium |
171 | Square Sum of the Digital Squares | For a positive integer $n$, let $f(n)$ be the sum of the squares of the digits (in base $10$) of $n$, e.g.
\begin{align}
f(3) &= 3^2 = 9,\\
f(25) &= 2^2 + 5^2 = 4 + 25 = 29,\\
f(442) &= 4^2 + 4^2 + 2^2 = 16 + 16 + 4 = 36\\
\end{align}
Find the last nine digits of the sum of all $n$, $0 \lt n \lt 10^{20}$, such that $f(... | For a positive integer $n$, let $f(n)$ be the sum of the squares of the digits (in base $10$) of $n$, e.g.
\begin{align}
f(3) &= 3^2 = 9,\\
f(25) &= 2^2 + 5^2 = 4 + 25 = 29,\\
f(442) &= 4^2 + 4^2 + 2^2 = 16 + 16 + 4 = 36\\
\end{align}
Find the last nine digits of the sum of all $n$, $0 \lt n \lt 10^{20}$, such that $f(... | <p>For a positive integer $n$, let $f(n)$ be the sum of the squares of the digits (in base $10$) of $n$, e.g.</p>
\begin{align}
f(3) &= 3^2 = 9,\\
f(25) &= 2^2 + 5^2 = 4 + 25 = 29,\\
f(442) &= 4^2 + 4^2 + 2^2 = 16 + 16 + 4 = 36\\
\end{align}
<p>Find the last nine digits of the sum of all $n$, $0 \lt n \lt 10^{20}$, suc... | 142989277 | Saturday, 8th December 2007, 05:00 am | 3134 | 65% | hard |
124 | Ordered Radicals | The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
If we calculate $\operatorname{rad}(n)$ for $1 \le n \le 10$, then sort them on $\operatorname{rad}(n)$, and sorting on... | The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
If we calculate $\operatorname{rad}(n)$ for $1 \le n \le 10$, then sort them on $\operatorname{rad}(n)$, and sorting on... | <p>The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.</p>
<p>If we calculate $\operatorname{rad}(n)$ for $1 \le n \le 10$, then sort them on $\operatorname{rad}(n)$, and ... | 21417 | Friday, 14th July 2006, 06:00 pm | 15011 | 25% | easy |
302 | Strong Achilles Numbers | A positive integer $n$ is powerful if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$.
A positive integer $n$ is a perfect power if $n$ can be expressed as a power of another positive integer.
A positive integer $n$ is an Achilles number if $n$ is powerful but not a perfect power. For example, $864$ and... | A positive integer $n$ is powerful if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$.
A positive integer $n$ is a perfect power if $n$ can be expressed as a power of another positive integer.
A positive integer $n$ is an Achilles number if $n$ is powerful but not a perfect power. For example, $864$ and... | <p>
A positive integer $n$ is <strong>powerful</strong> if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$.
</p>
<p>
A positive integer $n$ is a <strong>perfect power</strong> if $n$ can be expressed as a power of another positive integer.
</p>
<p>
A positive integer $n$ is an <strong>Achilles number</stron... | 1170060 | Saturday, 18th September 2010, 07:00 pm | 837 | 60% | hard |
616 | Creative Numbers | Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:
remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$
or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a, b > 1$, and add both $a$ and ... | Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:
remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$
or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a, b > 1$, and add both $a$ and ... | <p>Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:
</p><ul><li>remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$</li>
<li>or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a, b >... | 310884668312456458 | Saturday, 16th December 2017, 01:00 pm | 529 | 40% | medium |
603 | Substring Sums of Prime Concatenations | Let $S(n)$ be the sum of all contiguous integer-substrings that can be formed from the integer $n$. The substrings need not be distinct.
For example, $S(2024) = 2 + 0 + 2 + 4 + 20 + 02 + 24 + 202 + 024 + 2024 = 2304$.
Let $P(n)$ be the integer formed by concatenating the first $n$ primes together. For example, $P(7) =... | Let $S(n)$ be the sum of all contiguous integer-substrings that can be formed from the integer $n$. The substrings need not be distinct.
For example, $S(2024) = 2 + 0 + 2 + 4 + 20 + 02 + 24 + 202 + 024 + 2024 = 2304$.
Let $P(n)$ be the integer formed by concatenating the first $n$ primes together. For example, $P(7) =... | <p>Let $S(n)$ be the sum of all contiguous integer-substrings that can be formed from the integer $n$. The substrings need not be distinct. </p>
<p>For example, $S(2024) = 2 + 0 + 2 + 4 + 20 + 02 + 24 + 202 + 024 + 2024 = 2304$.</p>
<p>Let $P(n)$ be the integer formed by concatenating the first $n$ primes together. For... | 879476477 | Sunday, 14th May 2017, 01:00 am | 470 | 45% | medium |
712 | Exponent Difference | For any integer $n>0$ and prime number $p,$ define $\nu_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$.
Define $$D(n, m) = \sum_{p \text{ prime}} \left| \nu_p(n) - \nu_p(m)\right|.$$ For example, $D(14,24) = 4$.
Furthermore, define $$S(N) = \sum_{1 \le n, m \le N} D(n, m).$$ You are given $S(10) = ... | For any integer $n>0$ and prime number $p,$ define $\nu_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$.
Define $$D(n, m) = \sum_{p \text{ prime}} \left| \nu_p(n) - \nu_p(m)\right|.$$ For example, $D(14,24) = 4$.
Furthermore, define $$S(N) = \sum_{1 \le n, m \le N} D(n, m).$$ You are given $S(10) = ... | <p>
For any integer $n>0$ and prime number $p,$ define $\nu_p(n)$ as the greatest integer $r$ such that $p^r$ divides $n$.
</p>
<p>
Define $$D(n, m) = \sum_{p \text{ prime}} \left| \nu_p(n) - \nu_p(m)\right|.$$ For example, $D(14,24) = 4$.
</p>
<p>
Furthermore, define $$S(N) = \sum_{1 \le n, m \le N} D(n, m).$$ Yo... | 413876461 | Saturday, 18th April 2020, 11:00 pm | 554 | 25% | easy |
787 | Bézout's Game | Two players play a game with two piles of stones. They take alternating turns. If there are currently $a$ stones in the first pile and $b$ stones in the second, a turn consists of removing $c\geq 0$ stones from the first pile and $d\geq 0$ from the second in such a way that $ad-bc=\pm1$. The winner is the player who fi... | Two players play a game with two piles of stones. They take alternating turns. If there are currently $a$ stones in the first pile and $b$ stones in the second, a turn consists of removing $c\geq 0$ stones from the first pile and $d\geq 0$ from the second in such a way that $ad-bc=\pm1$. The winner is the player who fi... | <p>Two players play a game with two piles of stones. They take alternating turns. If there are currently $a$ stones in the first pile and $b$ stones in the second, a turn consists of removing $c\geq 0$ stones from the first pile and $d\geq 0$ from the second in such a way that $ad-bc=\pm1$. The winner is the player who... | 202642367520564145 | Saturday, 26th February 2022, 01:00 pm | 208 | 45% | medium |
843 | Periodic Circles | This problem involves an iterative procedure that begins with a circle of $n\ge 3$ integers. At each step every number is simultaneously replaced with the absolute difference of its two neighbours.
For any initial values, the procedure eventually becomes periodic.
Let $S(N)$ be the sum of all possible periods for $3\... | This problem involves an iterative procedure that begins with a circle of $n\ge 3$ integers. At each step every number is simultaneously replaced with the absolute difference of its two neighbours.
For any initial values, the procedure eventually becomes periodic.
Let $S(N)$ be the sum of all possible periods for $3\... | <p>
This problem involves an iterative procedure that begins with a circle of $n\ge 3$ integers. At each step every number is simultaneously replaced with the absolute difference of its two neighbours.</p>
<p>
For any initial values, the procedure eventually becomes periodic.</p>
<p>
Let $S(N)$ be the sum of all possib... | 2816775424692 | Sunday, 14th May 2023, 11:00 am | 133 | 80% | hard |
509 | Divisor Nim | Anton and Bertrand love to play three pile Nim.
However, after a lot of games of Nim they got bored and changed the rules somewhat.
They may only take a number of stones from a pile that is a proper divisora proper divisor of $n$ is a divisor of $n$ smaller than $n$ of the number of stones present in the pile. E.g. if ... | Anton and Bertrand love to play three pile Nim.
However, after a lot of games of Nim they got bored and changed the rules somewhat.
They may only take a number of stones from a pile that is a proper divisora proper divisor of $n$ is a divisor of $n$ smaller than $n$ of the number of stones present in the pile. E.g. if ... | <p>
Anton and Bertrand love to play three pile Nim.<br/>
However, after a lot of games of Nim they got bored and changed the rules somewhat.<br/>
They may only take a number of stones from a pile that is a <dfn class="tooltip">proper divisor<span class="tooltiptext">a proper divisor of $n$ is a divisor of $n$ smaller t... | 151725678 | Saturday, 28th March 2015, 01:00 pm | 664 | 45% | medium |
719 | Number Splitting | We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation of $n$ into $2$ or more numbers then adding the numbers.
For example, $81$ is an $S$-number because $\sqrt{81} = 8+1$.
$6724$ is an $S$-number: $\sqrt{6724} = 6+72+... | We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation of $n$ into $2$ or more numbers then adding the numbers.
For example, $81$ is an $S$-number because $\sqrt{81} = 8+1$.
$6724$ is an $S$-number: $\sqrt{6724} = 6+72+... | <p>
We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation of $n$ into $2$ or more numbers then adding the numbers.
</p>
<p>
For example, $81$ is an $S$-number because $\sqrt{81} = 8+1$.<br/>
$6724$ is an $S$-number: $\sq... | 128088830547982 | Saturday, 6th June 2020, 08:00 pm | 4460 | 5% | easy |
600 | Integer Sided Equiangular Hexagons | Let $H(n)$ be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding $n$.
Hexagons are distinct if and only if they are not congruent.
You are given $H(6) = 1$, $H(12) = 10$, $H(100) = 31248$.
Find $H(55106)$.
Equiangular hexagons with perimeter not exceeding $12$ | Let $H(n)$ be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding $n$.
Hexagons are distinct if and only if they are not congruent.
You are given $H(6) = 1$, $H(12) = 10$, $H(100) = 31248$.
Find $H(55106)$.
Equiangular hexagons with perimeter not exceeding $12$ | <p>Let $H(n)$ be the number of distinct integer sided <strong>equiangular</strong> convex hexagons with perimeter not exceeding $n$.<br/>
Hexagons are distinct if and only if they are not <strong>congruent</strong>.</p>
<p>You are given $H(6) = 1$, $H(12) = 10$, $H(100) = 31248$.<br/>
Find $H(55106)$.</p>
<div class="c... | 2668608479740672 | Saturday, 22nd April 2017, 04:00 pm | 673 | 35% | medium |
222 | Sphere Packing | What is the length of the shortest pipe, of internal radius $\pu{50 mm}$, that can fully contain $21$ balls of radii $\pu{30 mm}, \pu{31 mm}, \dots, \pu{50 mm}$?
Give your answer in micrometres ($\pu{10^{-6} m}$) rounded to the nearest integer. | What is the length of the shortest pipe, of internal radius $\pu{50 mm}$, that can fully contain $21$ balls of radii $\pu{30 mm}, \pu{31 mm}, \dots, \pu{50 mm}$?
Give your answer in micrometres ($\pu{10^{-6} m}$) rounded to the nearest integer. | <p>What is the length of the shortest pipe, of internal radius $\pu{50 mm}$, that can fully contain $21$ balls of radii $\pu{30 mm}, \pu{31 mm}, \dots, \pu{50 mm}$?</p>
<p>Give your answer in micrometres ($\pu{10^{-6} m}$) rounded to the nearest integer.</p> | 1590933 | Friday, 19th December 2008, 01:00 pm | 2310 | 60% | hard |
655 | Divisible Palindromes | The numbers $545$, $5\,995$ and $15\,151$ are the three smallest palindromes divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$.
How many palindromes less than $10^{32}$ are divisible by $10\,000\,019\,$ ? | The numbers $545$, $5\,995$ and $15\,151$ are the three smallest palindromes divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$.
How many palindromes less than $10^{32}$ are divisible by $10\,000\,019\,$ ? | <p>The numbers $545$, $5\,995$ and $15\,151$ are the three smallest <b>palindromes</b> divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$.</p>
<p>How many palindromes less than $10^{32}$ are divisible by $10\,000\,019\,$ ?</p> | 2000008332 | Sunday, 10th February 2019, 07:00 am | 610 | 30% | easy |
112 | Bouncy Numbers | Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.
We shall call a positive integer that is neither increasing nor decrea... | Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.
We shall call a positive integer that is neither increasing nor decrea... | <p>Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.</p>
<p>Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.</p>
<p>We shall call a positive integer that is neither incr... | 1587000 | Friday, 30th December 2005, 06:00 pm | 26945 | 15% | easy |
476 | Circle Packing II | Let $R(a, b, c)$ be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths $a$, $b$ and $c$.
Let $S(n)$ be the average value of $R(a, b, c)$ over all integer triplets $(a, b, c)$ such that $1 \le a \le b \le c \lt a + b \le n$.
You are given $S(2) = R(1, 1, 1) \approx 0.31998$, $S... | Let $R(a, b, c)$ be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths $a$, $b$ and $c$.
Let $S(n)$ be the average value of $R(a, b, c)$ over all integer triplets $(a, b, c)$ such that $1 \le a \le b \le c \lt a + b \le n$.
You are given $S(2) = R(1, 1, 1) \approx 0.31998$, $S... | <p>Let $R(a, b, c)$ be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths $a$, $b$ and $c$.</p>
<p>Let $S(n)$ be the average value of $R(a, b, c)$ over all integer triplets $(a, b, c)$ such that $1 \le a \le b \le c \lt a + b \le n$.</p>
<p>You are given $S(2) = R(1, 1, 1) \ap... | 110242.87794 | Saturday, 14th June 2014, 01:00 pm | 453 | 45% | medium |
485 | Maximum Number of Divisors | Let $d(n)$ be the number of divisors of $n$.
Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$.
Let $S(u,k)$ be the sum of $M(n,k)$ for $1 \le n \le u-k+1$.
You are given that $S(1000,10)=17176$.
Find $S(100\,000\,000, 100\,000)$. | Let $d(n)$ be the number of divisors of $n$.
Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$.
Let $S(u,k)$ be the sum of $M(n,k)$ for $1 \le n \le u-k+1$.
You are given that $S(1000,10)=17176$.
Find $S(100\,000\,000, 100\,000)$. | <p>
Let $d(n)$ be the number of divisors of $n$.<br/>
Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$.<br/>
Let $S(u,k)$ be the sum of $M(n,k)$ for $1 \le n \le u-k+1$.
</p>
<p>
You are given that $S(1000,10)=17176$.
</p>
<p>
Find $S(100\,000\,000, 100\,000)$.
</p> | 51281274340 | Saturday, 18th October 2014, 04:00 pm | 1293 | 30% | easy |
330 | Euler's Number | An infinite sequence of real numbers $a(n)$ is defined for all integers $n$ as follows:
$$a(n) = \begin{cases}
1 & n \lt 0\\
\sum \limits_{i = 1}^{\infty}{\dfrac{a(n - i)}{i!}} & n \ge 0
\end{cases}$$
For example,
$a(0) = \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots = e - 1$
$a(1) = \dfrac{e - 1}{1!} + \dfra... | An infinite sequence of real numbers $a(n)$ is defined for all integers $n$ as follows:
$$a(n) = \begin{cases}
1 & n \lt 0\\
\sum \limits_{i = 1}^{\infty}{\dfrac{a(n - i)}{i!}} & n \ge 0
\end{cases}$$
For example,
$a(0) = \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots = e - 1$
$a(1) = \dfrac{e - 1}{1!} + \dfra... | An infinite sequence of real numbers $a(n)$ is defined for all integers $n$ as follows:
$$a(n) = \begin{cases}
1 & n \lt 0\\
\sum \limits_{i = 1}^{\infty}{\dfrac{a(n - i)}{i!}} & n \ge 0
\end{cases}$$
<p>For example,<br/></p>
<p>$a(0) = \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots = e - 1$<br/>
$a(1) = \dfra... | 15955822 | Sunday, 27th March 2011, 05:00 am | 587 | 70% | hard |
280 | Ant and Seeds | A laborious ant walks randomly on a $5 \times 5$ grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are $2$, $3$ or $4$ possible moves at each step depending on the ant's position.
At the start of the walk, a seed is placed on... | A laborious ant walks randomly on a $5 \times 5$ grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are $2$, $3$ or $4$ possible moves at each step depending on the ant's position.
At the start of the walk, a seed is placed on... | <p>A laborious ant walks randomly on a $5 \times 5$ grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are $2$, $3$ or $4$ possible moves at each step depending on the ant's position.</p>
<p>At the start of the walk, a seed is... | 430.088247 | Saturday, 27th February 2010, 01:00 pm | 1169 | 65% | hard |
278 | Linear Combinations of Semiprimes | Given the values of integers $1 < a_1 < a_2 < \dots < a_n$, consider the linear combination
$q_1 a_1+q_2 a_2 + \dots + q_n a_n=b$, using only integer values $q_k \ge 0$.
Note that for a given set of $a_k$, it may be that not all values of $b$ are possible.
For instance, if $a_1=5$ and $a_2=7$, there are no $q_1 \ge ... | Given the values of integers $1 < a_1 < a_2 < \dots < a_n$, consider the linear combination
$q_1 a_1+q_2 a_2 + \dots + q_n a_n=b$, using only integer values $q_k \ge 0$.
Note that for a given set of $a_k$, it may be that not all values of $b$ are possible.
For instance, if $a_1=5$ and $a_2=7$, there are no $q_1 \ge ... | <p>
Given the values of integers $1 < a_1 < a_2 < \dots < a_n$, consider the linear combination<br>
$q_1 a_1+q_2 a_2 + \dots + q_n a_n=b$, using only integer values $q_k \ge 0$.
</br></p>
<p>
Note that for a given set of $a_k$, it may be that not all values of $b$ are possible.<br/>
For instance, if $a_1=5... | 1228215747273908452 | Saturday, 13th February 2010, 05:00 am | 1161 | 50% | medium |
350 | Constraining the Least Greatest and the Greatest Least | A list of size $n$ is a sequence of $n$ natural numbers. Examples are $(2,4,6)$, $(2,6,4)$, $(10,6,15,6)$, and $(11)$.
The greatest common divisor, or $\gcd$, of a list is the largest natural number that divides all entries of the list. Examples: $\gcd(2,6,4) = 2$, $\gcd(10,6,15,6) = 1$ and $\gcd(11) = 11$.
The least... | A list of size $n$ is a sequence of $n$ natural numbers. Examples are $(2,4,6)$, $(2,6,4)$, $(10,6,15,6)$, and $(11)$.
The greatest common divisor, or $\gcd$, of a list is the largest natural number that divides all entries of the list. Examples: $\gcd(2,6,4) = 2$, $\gcd(10,6,15,6) = 1$ and $\gcd(11) = 11$.
The least... | <p>A <dfn>list of size $n$</dfn> is a sequence of $n$ natural numbers.<br/> Examples are $(2,4,6)$, $(2,6,4)$, $(10,6,15,6)$, and $(11)$.
</p><p>
The <strong>greatest common divisor</strong>, or $\gcd$, of a list is the largest natural number that divides all entries of the list. <br/>Examples: $\gcd(2,6,4) = 2$, $\gcd... | 84664213 | Saturday, 10th September 2011, 07:00 pm | 531 | 60% | hard |
884 | Removing Cubes | Starting from a positive integer $n$, at each step we subtract from $n$ the largest perfect cube not exceeding $n$, until $n$ becomes $0$.
For example, with $n = 100$ the procedure ends in $4$ steps:
$$100 \xrightarrow{-4^3} 36 \xrightarrow{-3^3} 9 \xrightarrow{-2^3} 1 \xrightarrow{-1^3} 0.$$
Let $D(n)$ denote the numb... | Starting from a positive integer $n$, at each step we subtract from $n$ the largest perfect cube not exceeding $n$, until $n$ becomes $0$.
For example, with $n = 100$ the procedure ends in $4$ steps:
$$100 \xrightarrow{-4^3} 36 \xrightarrow{-3^3} 9 \xrightarrow{-2^3} 1 \xrightarrow{-1^3} 0.$$
Let $D(n)$ denote the numb... | <p>
Starting from a positive integer $n$, at each step we subtract from $n$ the largest perfect cube not exceeding $n$, until $n$ becomes $0$.<br/>
For example, with $n = 100$ the procedure ends in $4$ steps:
$$100 \xrightarrow{-4^3} 36 \xrightarrow{-3^3} 9 \xrightarrow{-2^3} 1 \xrightarrow{-1^3} 0.$$
Let $D(n)$ denote... | 1105985795684653500 | Sunday, 31st March 2024, 08:00 am | 588 | 20% | easy |
188 | Hyperexponentiation | The hyperexponentiation or tetration of a number $a$ by a positive integer $b$, denoted by $a\mathbin{\uparrow \uparrow}b$ or $^b a$, is recursively defined by:
$a \mathbin{\uparrow \uparrow} 1 = a$,
$a \mathbin{\uparrow \uparrow} (k+1) = a^{(a \mathbin{\uparrow \uparrow} k)}$.
Thus we have e.g. $3 \mathbin{\uparrow \... | The hyperexponentiation or tetration of a number $a$ by a positive integer $b$, denoted by $a\mathbin{\uparrow \uparrow}b$ or $^b a$, is recursively defined by:
$a \mathbin{\uparrow \uparrow} 1 = a$,
$a \mathbin{\uparrow \uparrow} (k+1) = a^{(a \mathbin{\uparrow \uparrow} k)}$.
Thus we have e.g. $3 \mathbin{\uparrow \... | <p>The <strong>hyperexponentiation</strong> or <strong>tetration</strong> of a number $a$ by a positive integer $b$, denoted by $a\mathbin{\uparrow \uparrow}b$ or $^b a$, is recursively defined by:<br/><br/>
$a \mathbin{\uparrow \uparrow} 1 = a$,<br/>
$a \mathbin{\uparrow \uparrow} (k+1) = a^{(a \mathbin{\uparrow \upar... | 95962097 | Friday, 4th April 2008, 02:00 pm | 6985 | 35% | medium |
66 | Diophantine Equation | Consider quadratic Diophantine equations of the form:
$$x^2 - Dy^2 = 1$$
For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.
It can be assumed that there are no solutions in positive integers when $D$ is square.
By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we obta... | Consider quadratic Diophantine equations of the form:
$$x^2 - Dy^2 = 1$$
For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.
It can be assumed that there are no solutions in positive integers when $D$ is square.
By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we obta... | <p>Consider quadratic Diophantine equations of the form:
$$x^2 - Dy^2 = 1$$</p>
<p>For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.</p>
<p>It can be assumed that there are no solutions in positive integers when $D$ is square.</p>
<p>By finding minimal solutions in $x$ for $D = \{2... | 661 | Friday, 26th March 2004, 06:00 pm | 22183 | 25% | easy |
902 | Permutation Powers | A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in one-line notation as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.
For example, $\text{rank}(2,1,3) = 3$ because the six permutations of $\{1, 2,... | A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in one-line notation as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.
For example, $\text{rank}(2,1,3) = 3$ because the six permutations of $\{1, 2,... | <p>A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in <b>one-line notation</b> as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.</p>
<p>For example, $\text{rank}(2,1,3) = 3$ because the six permuta... | 343557869 | Sunday, 28th July 2024, 11:00 am | 165 | 50% | medium |
612 | Friend Numbers | Let's call two numbers friend numbers if their representation in base $10$ has at least one common digit. E.g. $1123$ and $3981$ are friend numbers.
Let $f(n)$ be the number of pairs $(p,q)$ with $1\le p \lt q \lt n$ such that $p$ and $q$ are friend numbers.
$f(100)=1539$.
Find $f(10^{18}) \bmod 1000267129$. | Let's call two numbers friend numbers if their representation in base $10$ has at least one common digit. E.g. $1123$ and $3981$ are friend numbers.
Let $f(n)$ be the number of pairs $(p,q)$ with $1\le p \lt q \lt n$ such that $p$ and $q$ are friend numbers.
$f(100)=1539$.
Find $f(10^{18}) \bmod 1000267129$. | <p>
Let's call two numbers <dfn>friend numbers</dfn> if their representation in base $10$ has at least one common digit.<br/> E.g. $1123$ and $3981$ are friend numbers.
</p>
<p>
Let $f(n)$ be the number of pairs $(p,q)$ with $1\le p \lt q \lt n$ such that $p$ and $q$ are friend numbers.<br/>
$f(100)=1539$.
</p>
<p>
Fi... | 819963842 | Sunday, 22nd October 2017, 01:00 am | 758 | 30% | easy |
584 | Birthday Problem Revisited | A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet. On examining the drone, they uncovered a device that sought the answer for the so called "Birthday Problem". The description of the problem was as follows:
If people on... | A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet. On examining the drone, they uncovered a device that sought the answer for the so called "Birthday Problem". The description of the problem was as follows:
If people on... | <p>A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet. On examining the drone, they uncovered a device that sought the answer for the so called "<b>Birthday Problem</b>". The description of the problem was as follows:</p... | 32.83822408 | Saturday, 31st December 2016, 04:00 pm | 234 | 100% | hard |
875 | Quadruple Congruence | For a positive integer $n$ we define $q(n)$ to be the number of solutions to:
$$a_1^2+a_2^2+a_3^2+a_4^2 \equiv b_1^2+b_2^2+b_3^2+b_4^2 \pmod n$$
where $0 \leq a_i, b_i \lt n$. For example, $q(4)= 18432$.
Define $\displaystyle Q(n)=\sum_{i=1}^{n}q(i)$. You are given $Q(10)=18573381$.
Find $Q(12345678)$. Give your answ... | For a positive integer $n$ we define $q(n)$ to be the number of solutions to:
$$a_1^2+a_2^2+a_3^2+a_4^2 \equiv b_1^2+b_2^2+b_3^2+b_4^2 \pmod n$$
where $0 \leq a_i, b_i \lt n$. For example, $q(4)= 18432$.
Define $\displaystyle Q(n)=\sum_{i=1}^{n}q(i)$. You are given $Q(10)=18573381$.
Find $Q(12345678)$. Give your answ... | <p>
For a positive integer $n$ we define $q(n)$ to be the number of solutions to:</p>
$$a_1^2+a_2^2+a_3^2+a_4^2 \equiv b_1^2+b_2^2+b_3^2+b_4^2 \pmod n$$
<p>where $0 \leq a_i, b_i \lt n$. For example, $q(4)= 18432$.</p>
<p>
Define $\displaystyle Q(n)=\sum_{i=1}^{n}q(i)$. You are given $Q(10)=18573381$.</p>
<p>
Find $Q(1... | 79645946 | Sunday, 4th February 2024, 07:00 am | 218 | 35% | medium |
70 | Totient Permutation | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as $1, 2, 4, 5, 7$, and $8$, are all less than nine and relatively prime to nine, $\phi(9)=6$.The number $1$ is considere... | Euler's totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as $1, 2, 4, 5, 7$, and $8$, are all less than nine and relatively prime to nine, $\phi(9)=6$.The number $1$ is considere... | <p>Euler's totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively prime to $n$. For example, as $1, 2, 4, 5, 7$, and $8$, are all less than nine and relatively prime to nine, $\phi(9)=6$.<br/>The number $1$ is c... | 8319823 | Friday, 21st May 2004, 06:00 pm | 24976 | 20% | easy |
411 | Uphill Paths | Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i \bmod n, 3^i \bmod n)$ for $0 \leq i \leq 2n$. We will consider stations with the same coordinates as the same station.
We wish to form a path from $(0, 0)$ to $(n, n)$ such that the $x$ and $y$ coordinates never decrease.
Let ... | Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i \bmod n, 3^i \bmod n)$ for $0 \leq i \leq 2n$. We will consider stations with the same coordinates as the same station.
We wish to form a path from $(0, 0)$ to $(n, n)$ such that the $x$ and $y$ coordinates never decrease.
Let ... | <p>
Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i \bmod n, 3^i \bmod n)$ for $0 \leq i \leq 2n$. We will consider stations with the same coordinates as the same station.
</p><p>
We wish to form a path from $(0, 0)$ to $(n, n)$ such that the $x$ and $y$ coordinates never dec... | 9936352 | Saturday, 19th January 2013, 10:00 pm | 747 | 45% | medium |
690 | Tom and Jerry | Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$.
Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes.
Originally, Jerry is hiding in one of the mouseholes.
Every morning, Tom can check one (and only one) of the mouseholes. If Jerry happens to be hidi... | Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$.
Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes.
Originally, Jerry is hiding in one of the mouseholes.
Every morning, Tom can check one (and only one) of the mouseholes. If Jerry happens to be hidi... | <p>
Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$.
</p>
<p>
Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes.
</p>
<p>
Originally, Jerry is hiding in one of the mouseholes.<br/>
Every morning, Tom can check one (and only one) of the mouseholes. If J... | 415157690 | Sunday, 24th November 2019, 07:00 am | 233 | 60% | hard |
638 | Weighted Lattice Paths | Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties:
The path begins at $(0,0)$ and ends at $(a,b)$.
The path consists only of unit moves upwards or to the right; that is the coordinates are increasing with every move.
Denote $A(P_{a,b})$ to be the area under the path. For the example... | Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties:
The path begins at $(0,0)$ and ends at $(a,b)$.
The path consists only of unit moves upwards or to the right; that is the coordinates are increasing with every move.
Denote $A(P_{a,b})$ to be the area under the path. For the example... | Let $P_{a,b}$ denote a path in a $a\times b$ lattice grid with following properties:
<ul>
<li>The path begins at $(0,0)$ and ends at $(a,b)$.</li>
<li>The path consists only of unit moves upwards or to the right; that is the coordinates are increasing with every move.</li>
</ul>
Denote $A(P_{a,b})$ to be the area unde... | 18423394 | Sunday, 7th October 2018, 04:00 am | 417 | 40% | medium |
178 | Step Numbers | Consider the number $45656$.
It can be seen that each pair of consecutive digits of $45656$ has a difference of one.
A number for which every pair of consecutive digits has a difference of one is called a step number.
A pandigital number contains every decimal digit from $0$ to $9$ at least once.
How many pandigital... | Consider the number $45656$.
It can be seen that each pair of consecutive digits of $45656$ has a difference of one.
A number for which every pair of consecutive digits has a difference of one is called a step number.
A pandigital number contains every decimal digit from $0$ to $9$ at least once.
How many pandigital... | Consider the number $45656$. <br/>
It can be seen that each pair of consecutive digits of $45656$ has a difference of one.<br/>
A number for which every pair of consecutive digits has a difference of one is called a step number.<br/>
A pandigital number contains every decimal digit from $0$ to $9$ at least once.<br/>
... | 126461847755 | Saturday, 19th January 2008, 01:00 am | 3806 | 55% | medium |
737 | Coin Loops | A game is played with many identical, round coins on a flat table.
Consider a line perpendicular to the table.
The first coin is placed on the table touching the line.
Then, one by one, the coins are placed horizontally on top of the previous coin and touching the line.
The complete stack of coins must be balanced af... | A game is played with many identical, round coins on a flat table.
Consider a line perpendicular to the table.
The first coin is placed on the table touching the line.
Then, one by one, the coins are placed horizontally on top of the previous coin and touching the line.
The complete stack of coins must be balanced af... | <p>
A game is played with many identical, round coins on a flat table.
</p>
<p>
Consider a line perpendicular to the table.<br>
The first coin is placed on the table touching the line.<br/>
Then, one by one, the coins are placed horizontally on top of the previous coin and touching the line.<br/>
The complete stack of ... | 757794899 | Sunday, 6th December 2020, 04:00 am | 405 | 30% | easy |
912 | Where are the Odds? | Let $s_n$ be the $n$-th positive integer that does not contain three consecutive ones in its binary representation.
For example, $s_1 = 1$ and $s_7 = 8$.
Define $F(N)$ to be the sum of $n^2$ for all $n\leq N$ where $s_n$ is odd. You are given $F(10)=199$.
Find $F(10^{16})$ giving your answer modulo $10^9+7$. | Let $s_n$ be the $n$-th positive integer that does not contain three consecutive ones in its binary representation.
For example, $s_1 = 1$ and $s_7 = 8$.
Define $F(N)$ to be the sum of $n^2$ for all $n\leq N$ where $s_n$ is odd. You are given $F(10)=199$.
Find $F(10^{16})$ giving your answer modulo $10^9+7$. | <p>
Let $s_n$ be the $n$-th positive integer that does not contain three consecutive ones in its binary representation.<br/>
For example, $s_1 = 1$ and $s_7 = 8$.
</p>
<p>
Define $F(N)$ to be the sum of $n^2$ for all $n\leq N$ where $s_n$ is odd. You are given $F(10)=199$.
</p>
<p>
Find $F(10^{16})$ giving your answer ... | 674045136 | Sunday, 13th October 2024, 11:00 am | 192 | 50% | medium |
118 | Pandigital Prime Sets | Using all of the digits $1$ through $9$ and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set $\{2,5,47,89,631\}$, all of the elements belonging to it are prime.
How many distinct sets containing each of the digits one through nine exactly once contain only pri... | Using all of the digits $1$ through $9$ and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set $\{2,5,47,89,631\}$, all of the elements belonging to it are prime.
How many distinct sets containing each of the digits one through nine exactly once contain only pri... | <p>Using all of the digits $1$ through $9$ and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set $\{2,5,47,89,631\}$, all of the elements belonging to it are prime.</p>
<p>How many distinct sets containing each of the digits one through nine exactly once contai... | 44680 | Friday, 24th March 2006, 06:00 pm | 7907 | 45% | medium |
587 | Concave Triangle | A square is drawn around a circle as shown in the diagram below on the left.
We shall call the blue shaded region the L-section.
A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right.
We shall call the orange shaded region a concave triangle.
It should be clear th... | A square is drawn around a circle as shown in the diagram below on the left.
We shall call the blue shaded region the L-section.
A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right.
We shall call the orange shaded region a concave triangle.
It should be clear th... | <p>
A square is drawn around a circle as shown in the diagram below on the left.<br/>
We shall call the blue shaded region the L-section.<br/>
A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right.<br/>
We shall call the orange shaded region a <dfn>concave triangle</df... | 2240 | Sunday, 22nd January 2017, 01:00 am | 3471 | 20% | easy |
373 | Circumscribed Circles | Every triangle has a circumscribed circle that goes through the three vertices.
Consider all integer sided triangles for which the radius of the circumscribed circle is integral as well.
Let $S(n)$ be the sum of the radii of the circumscribed circles of all such triangles for which the radius does not exceed $n$.
$S... | Every triangle has a circumscribed circle that goes through the three vertices.
Consider all integer sided triangles for which the radius of the circumscribed circle is integral as well.
Let $S(n)$ be the sum of the radii of the circumscribed circles of all such triangles for which the radius does not exceed $n$.
$S... | <p>
Every triangle has a circumscribed circle that goes through the three vertices.
Consider all integer sided triangles for which the radius of the circumscribed circle is integral as well.
</p>
<p>
Let $S(n)$ be the sum of the radii of the circumscribed circles of all such triangles for which the radius does not exce... | 727227472448913 | Saturday, 25th February 2012, 04:00 pm | 387 | 75% | hard |
11 | Largest Product in a Grid | In the $20 \times 20$ grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 ... | In the $20 \times 20$ grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 ... | <p>In the $20 \times 20$ grid below, four numbers along a diagonal line have been marked in red.</p>
<p class="monospace center">
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08<br/>
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00<br/>
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65<... | 70600674 | Friday, 22nd February 2002, 06:00 pm | 252014 | 5% | easy |
629 | Scatterstone Nim | Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary set of stone piles with a total number of stones equal to $n$.
During a player's turn, he/she must pick a pile having at least $2$ stones and perform a split op... | Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary set of stone piles with a total number of stones equal to $n$.
During a player's turn, he/she must pick a pile having at least $2$ stones and perform a split op... | <p>Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary set of stone piles with a total number of stones equal to $n$.</p>
<p>During a player's turn, he/she must pick a pile having at least $2$ stones and perform ... | 626616617 | Sunday, 17th June 2018, 04:00 am | 272 | 55% | medium |
853 | Pisano Periods 1 | For every positive integer $n$ the Fibonacci sequence modulo
$n$ is periodic. The period depends on the value of $n$.
This period is called the Pisano period for $n$, often shortened to $\pi(n)$.
There are three values of $n$ for which
$\pi(n)$ equals $18$: $19$, $38$ and $76$. The sum of those smaller than $50$ is ... | For every positive integer $n$ the Fibonacci sequence modulo
$n$ is periodic. The period depends on the value of $n$.
This period is called the Pisano period for $n$, often shortened to $\pi(n)$.
There are three values of $n$ for which
$\pi(n)$ equals $18$: $19$, $38$ and $76$. The sum of those smaller than $50$ is ... | <p>
For every positive integer $n$ the Fibonacci sequence modulo
$n$ is periodic. The period depends on the value of $n$.
This period is called the <strong>Pisano period</strong> for $n$, often shortened to $\pi(n)$.</p>
<p>
There are three values of $n$ for which
$\pi(n)$ equals $18$: $19$, $38$ and $76$. The sum of... | 44511058204 | Saturday, 9th September 2023, 05:00 pm | 1469 | 5% | easy |
670 | Colouring a Strip | A certain type of tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a $2\times n$ rectangle, where $n$ is a positi... | A certain type of tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a $2\times n$ rectangle, where $n$ is a positi... | <p>A certain type of tile comes in three different sizes - $1 \times 1$, $1 \times 2$, and $1 \times 3$ - and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.</p>
<p>These are used to tile a $2\times n$ rectangle, where $n$ i... | 551055065 | Sunday, 19th May 2019, 01:00 am | 375 | 40% | medium |
64 | Odd Period Square Roots | All square roots are periodic when written as continued fractions and can be written in the form:
$\displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1 {a_3+ \dots}}}$
For example, let us consider $\sqrt{23}:$
$\quad \quad \sqrt{23}=4+\sqrt{23}-4=4+\frac 1 {\frac 1 {\sqrt{23}-4}}=4+\frac 1 {1+\f... | All square roots are periodic when written as continued fractions and can be written in the form:
$\displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1 {a_3+ \dots}}}$
For example, let us consider $\sqrt{23}:$
$\quad \quad \sqrt{23}=4+\sqrt{23}-4=4+\frac 1 {\frac 1 {\sqrt{23}-4}}=4+\frac 1 {1+\f... | <p>All square roots are periodic when written as continued fractions and can be written in the form:</p>
$\displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1 {a_3+ \dots}}}$
<p>For example, let us consider $\sqrt{23}:$</p>
$\quad \quad \sqrt{23}=4+\sqrt{23}-4=4+\frac 1 {\frac 1 {\sqrt{23}-4}}=4+... | 1322 | Friday, 27th February 2004, 06:00 pm | 24796 | 20% | easy |
736 | Paths to Equality | Define two functions on lattice points:
$r(x,y) = (x+1,2y)$
$s(x,y) = (2x,y+1)$
A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$, where:
$(a_1,b_1) = (a,b)$
$(a_k,b_k) = r(a_{k-1},b_{k-1})$ or $(a_k,b_k) = s(a_{k-1},b_{k-1})$ for $k > 1$
$a_k \ne b_k$ fo... | Define two functions on lattice points:
$r(x,y) = (x+1,2y)$
$s(x,y) = (2x,y+1)$
A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$, where:
$(a_1,b_1) = (a,b)$
$(a_k,b_k) = r(a_{k-1},b_{k-1})$ or $(a_k,b_k) = s(a_{k-1},b_{k-1})$ for $k > 1$
$a_k \ne b_k$ fo... | <p>Define two functions on lattice points:</p>
<center>$r(x,y) = (x+1,2y)$</center>
<center>$s(x,y) = (2x,y+1)$</center>
<p>A <i>path to equality</i> of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$, where:</p>
<ul><li>$(a_1,b_1) = (a,b)$</li>
<li>$(a_k,b_k) = r(a_{k-1},b_... | 25332747903959376 | Sunday, 29th November 2020, 01:00 am | 243 | 50% | medium |
501 | Eight Divisors | The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$.
The ten numbers not exceeding $100$ having exactly eight divisors are $24, 30, 40, 42, 54, 56, 66, 70, 78$ and $88$.
Let $f(n)$ be the count of numbers not exceeding $n$ with exactly eight divisors.
You are given $f(100) = 10$, $f(1000) = 180$ and $f(10^6)... | The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$.
The ten numbers not exceeding $100$ having exactly eight divisors are $24, 30, 40, 42, 54, 56, 66, 70, 78$ and $88$.
Let $f(n)$ be the count of numbers not exceeding $n$ with exactly eight divisors.
You are given $f(100) = 10$, $f(1000) = 180$ and $f(10^6)... | <p>The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$.
The ten numbers not exceeding $100$ having exactly eight divisors are $24, 30, 40, 42, 54, 56, 66, 70, 78$ and $88$.
Let $f(n)$ be the count of numbers not exceeding $n$ with exactly eight divisors.<br/>
You are given $f(100) = 10$, $f(1000) = 180$ and ... | 197912312715 | Saturday, 31st January 2015, 01:00 pm | 1521 | 40% | medium |
2 | Even Fibonacci Numbers | Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with $1$ and $2$, the first $10$ terms will be:
$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$$
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued term... | Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with $1$ and $2$, the first $10$ terms will be:
$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$$
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued term... | <p>Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with $1$ and $2$, the first $10$ terms will be:
$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$$</p>
<p>By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-v... | 4613732 | Friday, 19th October 2001, 06:00 pm | 816741 | 5% | easy |
43 | Sub-string Divisibility | The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.
Let $d_1$ be the $1$st digit, $d_2$ be the $2$nd digit, and so on. In this way, we note the following:
$d_2d_3d_4=406$ ... | The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.
Let $d_1$ be the $1$st digit, $d_2$ be the $2$nd digit, and so on. In this way, we note the following:
$d_2d_3d_4=406$ ... | <p>The number, $1406357289$, is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.</p>
<p>Let $d_1$ be the $1$<sup>st</sup> digit, $d_2$ be the $2$<sup>nd</sup> digit, and so on. In this way, we note... | 16695334890 | Friday, 9th May 2003, 06:00 pm | 65767 | 5% | easy |
635 | Subset Sums | Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, ..., q \cdot n\}$ that satisfy two conditions:
1) $B$ has exactly $n$ elements;
2) the sum of the elements of $B$ is divisible by $n$.
E.g. $A_2(5)=52$ and $A_3(5)=603$.
Let $S_q(L)$ be $\sum A_q(p)$ where the sum is taken over all primes $p \le L$.
E.g... | Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, ..., q \cdot n\}$ that satisfy two conditions:
1) $B$ has exactly $n$ elements;
2) the sum of the elements of $B$ is divisible by $n$.
E.g. $A_2(5)=52$ and $A_3(5)=603$.
Let $S_q(L)$ be $\sum A_q(p)$ where the sum is taken over all primes $p \le L$.
E.g... | <p>
Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, ..., q \cdot n\}$ that satisfy two conditions:<br/>
1) $B$ has exactly $n$ elements;<br/>
2) the sum of the elements of $B$ is divisible by $n$.
</p>
<p>
E.g. $A_2(5)=52$ and $A_3(5)=603$.
</p>
Let $S_q(L)$ be $\sum A_q(p)$ where the sum is taken over ... | 689294705 | Saturday, 25th August 2018, 07:00 pm | 387 | 40% | medium |
530 | GCD of Divisors | Every divisor $d$ of a number $n$ has a complementary divisor $n/d$.
Let $f(n)$ be the sum of the greatest common divisor of $d$ and $n/d$ over all positive divisors $d$ of $n$, that is
$f(n)=\displaystyle\sum_{d\mid n}\gcd(d,\frac n d)$.
Let $F$ be the summatory function of $f$, that is
$F(k)=\displaystyle\sum_{n=1}^k... | Every divisor $d$ of a number $n$ has a complementary divisor $n/d$.
Let $f(n)$ be the sum of the greatest common divisor of $d$ and $n/d$ over all positive divisors $d$ of $n$, that is
$f(n)=\displaystyle\sum_{d\mid n}\gcd(d,\frac n d)$.
Let $F$ be the summatory function of $f$, that is
$F(k)=\displaystyle\sum_{n=1}^k... | <p>Every divisor $d$ of a number $n$ has a <strong>complementary divisor</strong> $n/d$.</p>
<p>Let $f(n)$ be the sum of the <strong>greatest common divisor</strong> of $d$ and $n/d$ over all positive divisors $d$ of $n$, that is
$f(n)=\displaystyle\sum_{d\mid n}\gcd(d,\frac n d)$.</p>
<p>Let $F$ be the summatory funct... | 207366437157977206 | Sunday, 18th October 2015, 01:00 am | 507 | 60% | hard |
168 | Number Rotations | Consider the number $142857$. We can right-rotate this number by moving the last digit ($7$) to the front of it, giving us $714285$.
It can be verified that $714285 = 5 \times 142857$.
This demonstrates an unusual property of $142857$: it is a divisor of its right-rotation.
Find the last $5$ digits of the sum of all in... | Consider the number $142857$. We can right-rotate this number by moving the last digit ($7$) to the front of it, giving us $714285$.
It can be verified that $714285 = 5 \times 142857$.
This demonstrates an unusual property of $142857$: it is a divisor of its right-rotation.
Find the last $5$ digits of the sum of all in... | <p>Consider the number $142857$. We can right-rotate this number by moving the last digit ($7$) to the front of it, giving us $714285$.<br/>
It can be verified that $714285 = 5 \times 142857$.<br/>
This demonstrates an unusual property of $142857$: it is a divisor of its right-rotation.</p>
<p>Find the last $5$ digits ... | 59206 | Friday, 16th November 2007, 05:00 pm | 2992 | 65% | hard |
810 | XOR-Primes | We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.
Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.
For example, $7 \otimes 3 = 9$, or in base $2$, $111_2 \otimes 11_2 ... | We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.
Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.
For example, $7 \otimes 3 = 9$, or in base $2$, $111_2 \otimes 11_2 ... | <p>We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.</p>
<p>Define the <dfn>XOR-product</dfn> of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.</p>
<p>For example, $7 \otimes 3 = 9$, or in ba... | 124136381 | Sunday, 2nd October 2022, 11:00 am | 794 | 20% | easy |
822 | Square the Smallest | A list initially contains the numbers $2, 3, \dots, n$.
At each round, the smallest number in the list is replaced by its square. If there is more than one such number, then only one of them is replaced.
For example, below are the first three rounds for $n = 5$:
$$[2, 3, 4, 5] \xrightarrow{(1)} [4, 3, 4, 5] \xrightar... | A list initially contains the numbers $2, 3, \dots, n$.
At each round, the smallest number in the list is replaced by its square. If there is more than one such number, then only one of them is replaced.
For example, below are the first three rounds for $n = 5$:
$$[2, 3, 4, 5] \xrightarrow{(1)} [4, 3, 4, 5] \xrightar... | <p>
A list initially contains the numbers $2, 3, \dots, n$.<br>
At each round, the smallest number in the list is replaced by its square. If there is more than one such number, then only one of them is replaced.
</br></p>
<p>
For example, below are the first three rounds for $n = 5$:
$$[2, 3, 4, 5] \xrightarrow{(1)} [4... | 950591530 | Saturday, 24th December 2022, 10:00 pm | 876 | 15% | easy |
77 | Prime Summations | It is possible to write ten as the sum of primes in exactly five different ways:
\begin{align}
&7 + 3\\
&5 + 5\\
&5 + 3 + 2\\
&3 + 3 + 2 + 2\\
&2 + 2 + 2 + 2 + 2
\end{align}
What is the first value which can be written as the sum of primes in over five thousand different ways? | It is possible to write ten as the sum of primes in exactly five different ways:
\begin{align}
&7 + 3\\
&5 + 5\\
&5 + 3 + 2\\
&3 + 3 + 2 + 2\\
&2 + 2 + 2 + 2 + 2
\end{align}
What is the first value which can be written as the sum of primes in over five thousand different ways? | <p>It is possible to write ten as the sum of primes in exactly five different ways:</p>
\begin{align}
&7 + 3\\
&5 + 5\\
&5 + 3 + 2\\
&3 + 3 + 2 + 2\\
&2 + 2 + 2 + 2 + 2
\end{align}
<p>What is the first value which can be written as the sum of primes in over five thousand different ways?</p> | 71 | Friday, 27th August 2004, 06:00 pm | 21578 | 25% | easy |
314 | The Mouse on the Moon | The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a $\pu{500 m}$ by $\pu{500 m}$ square area, but they will possess only that area which they wal... | The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a $\pu{500 m}$ by $\pu{500 m}$ square area, but they will possess only that area which they wal... | <p>
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a $\pu{500 m}$ by $\pu{500 m}$ square area, but they will possess only that area which they... | 132.52756426 | Sunday, 12th December 2010, 07:00 am | 579 | 80% | hard |
297 | Zeckendorf Representation | Each new term in the Fibonacci sequence is generated by adding the previous two terms.
Starting with $1$ and $2$, the first $10$ terms will be: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$.
Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, $100 = 3 + 8 + 89$.... | Each new term in the Fibonacci sequence is generated by adding the previous two terms.
Starting with $1$ and $2$, the first $10$ terms will be: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$.
Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, $100 = 3 + 8 + 89$.... | <p>Each new term in the Fibonacci sequence is generated by adding the previous two terms.<br/>
Starting with $1$ and $2$, the first $10$ terms will be: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$.</p>
<p>Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, $100... | 2252639041804718029 | Friday, 18th June 2010, 05:00 pm | 3045 | 35% | medium |
503 | Compromise or Persist | Alice is playing a game with $n$ cards numbered $1$ to $n$.
A game consists of iterations of the following steps.
(1) Alice picks one of the cards at random.
(2) Alice cannot see the number on it. Instead, Bob, one of her friends, sees the number and tells Alice how many previously-seen numbers are bigger than the numb... | Alice is playing a game with $n$ cards numbered $1$ to $n$.
A game consists of iterations of the following steps.
(1) Alice picks one of the cards at random.
(2) Alice cannot see the number on it. Instead, Bob, one of her friends, sees the number and tells Alice how many previously-seen numbers are bigger than the numb... | <p>Alice is playing a game with $n$ cards numbered $1$ to $n$.</p>
<p>A game consists of iterations of the following steps.<br/>
(1) Alice picks one of the cards at random.<br/>
(2) Alice cannot see the number on it. Instead, Bob, one of her friends, sees the number and tells Alice how many previously-seen numbers are ... | 3.8694550145 | Saturday, 14th February 2015, 07:00 pm | 359 | 60% | hard |
491 | Double Pandigital Number Divisible by $11$ | We call a positive integer double pandigital if it uses all the digits $0$ to $9$ exactly twice (with no leading zero). For example, $40561817703823564929$ is one such number.
How many double pandigital numbers are divisible by $11$? | We call a positive integer double pandigital if it uses all the digits $0$ to $9$ exactly twice (with no leading zero). For example, $40561817703823564929$ is one such number.
How many double pandigital numbers are divisible by $11$? | <p>We call a positive integer <dfn>double pandigital</dfn> if it uses all the digits $0$ to $9$ exactly twice (with no leading zero). For example, $40561817703823564929$ is one such number.</p>
<p>How many double pandigital numbers are divisible by $11$?</p> | 194505988824000 | Sunday, 30th November 2014, 10:00 am | 2368 | 20% | easy |
721 | High Powers of Irrational Numbers | Given is the function $f(a,n)=\lfloor (\lceil \sqrt a \rceil + \sqrt a)^n \rfloor$.
$\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.
$f(5,2)=27$ and $f(5,5)=3935$.
$G(n) = \displaystyle \sum_{a=1}^n f(a, a^2).$
$G(1000) \bmod 999\,999\,937=163861845. $
Find $G... | Given is the function $f(a,n)=\lfloor (\lceil \sqrt a \rceil + \sqrt a)^n \rfloor$.
$\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.
$f(5,2)=27$ and $f(5,5)=3935$.
$G(n) = \displaystyle \sum_{a=1}^n f(a, a^2).$
$G(1000) \bmod 999\,999\,937=163861845. $
Find $G... | <p>
Given is the function $f(a,n)=\lfloor (\lceil \sqrt a \rceil + \sqrt a)^n \rfloor$.<br/>
$\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.<br/>
$f(5,2)=27$ and $f(5,5)=3935$.
</p>
<p>
$G(n) = \displaystyle \sum_{a=1}^n f(a, a^2).$<br/>
$G(1000) \bmod 999\,999... | 700792959 | Sunday, 21st June 2020, 02:00 am | 472 | 30% | easy |
838 | Not Coprime | Let $f(N)$ be the smallest positive integer that is not coprime to any positive integer $n \le N$ whose least significant digit is $3$.
For example $f(40)$ equals to $897 = 3 \cdot 13 \cdot 23$ since it is not coprime to any of $3,13,23,33$. By taking the natural logarithm (log to base $e$) we obtain $\ln f(40) = \ln 8... | Let $f(N)$ be the smallest positive integer that is not coprime to any positive integer $n \le N$ whose least significant digit is $3$.
For example $f(40)$ equals to $897 = 3 \cdot 13 \cdot 23$ since it is not coprime to any of $3,13,23,33$. By taking the natural logarithm (log to base $e$) we obtain $\ln f(40) = \ln 8... | <p>Let $f(N)$ be the smallest positive integer that is not coprime to any positive integer $n \le N$ whose least significant digit is $3$.</p>
<p>For example $f(40)$ equals to $897 = 3 \cdot 13 \cdot 23$ since it is not coprime to any of $3,13,23,33$. By taking the <b><a href="https://en.wikipedia.org/wiki/Natural_loga... | 250591.442792 | Saturday, 8th April 2023, 08:00 pm | 640 | 20% | easy |
544 | Chromatic Conundrum | Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.
For example, $F(2,2,3) = 18$, $F(2,2,20) = 130340$, and $F(3,4,6) = 102923... | Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.
For example, $F(2,2,3) = 18$, $F(2,2,20) = 130340$, and $F(3,4,6) = 102923... | <p>Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.</p>
<p>For example, $F(2,2,3) = 18$, $F(2,2,20) = 130340$, and $F(3,4,6... | 640432376 | Saturday, 23rd January 2016, 07:00 pm | 291 | 90% | hard |
642 | Sum of Largest Prime Factors | Let $f(n)$ be the largest prime factor of $n$ and $\displaystyle F(n) = \sum_{i=2}^n f(i)$.
For example $F(10)=32$, $F(100)=1915$ and $F(10000)=10118280$.
Find $F(201820182018)$. Give your answer modulus $10^9$. | Let $f(n)$ be the largest prime factor of $n$ and $\displaystyle F(n) = \sum_{i=2}^n f(i)$.
For example $F(10)=32$, $F(100)=1915$ and $F(10000)=10118280$.
Find $F(201820182018)$. Give your answer modulus $10^9$. | <p>Let $f(n)$ be the largest prime factor of $n$ and $\displaystyle F(n) = \sum_{i=2}^n f(i)$.<br/>
For example $F(10)=32$, $F(100)=1915$ and $F(10000)=10118280$.</p>
<p>
Find $F(201820182018)$. Give your answer modulus $10^9$.</p> | 631499044 | Saturday, 10th November 2018, 04:00 pm | 402 | 45% | medium |
866 | Tidying Up B | A small child has a “number caterpillar” consisting of $N$ jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $N$ in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up... | A small child has a “number caterpillar” consisting of $N$ jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $N$ in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up... | <p>
A small child has a “number caterpillar” consisting of $N$ jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $N$ in order.</p>
<p>
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. ... | 492401720 | Sunday, 3rd December 2023, 04:00 am | 423 | 20% | easy |
618 | Numbers with a Given Prime Factor Sum | Consider the numbers $15$, $16$ and $18$:
$15=3\times 5$ and $3+5=8$.
$16 = 2\times 2\times 2\times 2$ and $2+2+2+2=8$.
$18 = 2\times 3\times 3$ and $2+3+3=8$.
$15$, $16$ and $18$ are the only numbers that have $8$ as sum of the prime factors (counted with multiplicity).
We define $S(k)$ to be the sum of all numbers... | Consider the numbers $15$, $16$ and $18$:
$15=3\times 5$ and $3+5=8$.
$16 = 2\times 2\times 2\times 2$ and $2+2+2+2=8$.
$18 = 2\times 3\times 3$ and $2+3+3=8$.
$15$, $16$ and $18$ are the only numbers that have $8$ as sum of the prime factors (counted with multiplicity).
We define $S(k)$ to be the sum of all numbers... | <p>Consider the numbers $15$, $16$ and $18$:<br/>
$15=3\times 5$ and $3+5=8$.<br/>
$16 = 2\times 2\times 2\times 2$ and $2+2+2+2=8$.<br/>
$18 = 2\times 3\times 3$ and $2+3+3=8$.<br/>
$15$, $16$ and $18$ are the only numbers that have $8$ as sum of the prime factors (counted with multiplicity).</p>
<p>
We define $S(k)... | 634212216 | Saturday, 13th January 2018, 07:00 pm | 1154 | 20% | easy |
291 | Panaitopol Primes | A prime number $p$ is called a Panaitopol prime if $p = \dfrac{x^4 - y^4}{x^3 + y^3}$ for some positive integers $x$ and $y$.
Find how many Panaitopol primes are less than $5 \times 10^{15}$. | A prime number $p$ is called a Panaitopol prime if $p = \dfrac{x^4 - y^4}{x^3 + y^3}$ for some positive integers $x$ and $y$.
Find how many Panaitopol primes are less than $5 \times 10^{15}$. | <p>
A prime number $p$ is called a Panaitopol prime if $p = \dfrac{x^4 - y^4}{x^3 + y^3}$ for some positive integers $x$ and $y$.</p>
<p>
Find how many Panaitopol primes are less than $5 \times 10^{15}$.
</p> | 4037526 | Friday, 7th May 2010, 09:00 pm | 1631 | 45% | medium |
780 | Toriangulations | For positive real numbers $a,b$, an $a\times b$ torus is a rectangle of width $a$ and height $b$, with left and right sides identified, as well as top and bottom sides identified. In other words, when tracing a path on the rectangle, reaching an edge results in "wrapping round" to the corresponding point on the opposit... | For positive real numbers $a,b$, an $a\times b$ torus is a rectangle of width $a$ and height $b$, with left and right sides identified, as well as top and bottom sides identified. In other words, when tracing a path on the rectangle, reaching an edge results in "wrapping round" to the corresponding point on the opposit... | <p>For positive real numbers $a,b$, an $a\times b$ <strong>torus</strong> is a rectangle of width $a$ and height $b$, with left and right sides identified, as well as top and bottom sides identified. In other words, when tracing a path on the rectangle, reaching an edge results in "wrapping round" to the corresponding ... | 613979935 | Saturday, 8th January 2022, 04:00 pm | 143 | 100% | hard |
213 | Flea Circus | A $30 \times 30$ grid of squares contains $900$ fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually $4$ possibilities, except for fleas on the edge of the grid or at the corners).
What is the expected number of unoccupied squares after $50$ rings of the be... | A $30 \times 30$ grid of squares contains $900$ fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually $4$ possibilities, except for fleas on the edge of the grid or at the corners).
What is the expected number of unoccupied squares after $50$ rings of the be... | <p>A $30 \times 30$ grid of squares contains $900$ fleas, initially one flea per square.<br/>
When a bell is rung, each flea jumps to an adjacent square at random (usually $4$ possibilities, except for fleas on the edge of the grid or at the corners).</p>
<p>What is the expected number of unoccupied squares after $50$ ... | 330.721154 | Saturday, 18th October 2008, 10:00 am | 2617 | 60% | hard |
556 | Squarefree Gaussian Integers | A Gaussian integer is a number $z = a + bi$ where $a$, $b$ are integers and $i^2 = -1$.
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which $b = 0$.
A Gaussian integer unit is one for which $a^2 + b^2 = 1$, i.e. one of $1, i, -1, -i$.
Let's define a prop... | A Gaussian integer is a number $z = a + bi$ where $a$, $b$ are integers and $i^2 = -1$.
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which $b = 0$.
A Gaussian integer unit is one for which $a^2 + b^2 = 1$, i.e. one of $1, i, -1, -i$.
Let's define a prop... | <p>A <b>Gaussian integer</b> is a number $z = a + bi$ where $a$, $b$ are integers and $i^2 = -1$.<br/>
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which $b = 0$.</p>
<p>A Gaussian integer <strong>unit</strong> is one for which $a^2 + b^2 = 1$, i.e. one... | 52126939292957 | Sunday, 17th April 2016, 07:00 am | 278 | 85% | hard |
894 | Spiral of Circles | Consider a unit circlecircle with radius 1 $C_0$ on the plane that does not enclose the origin. For $k\ge 1$, a circle $C_k$ is created by scaling and rotating $C_{k - 1}$ with respect to the origin. That is, both the radius and the distance to the origin are scaled by the same factor, and the centre of rotation is the... | Consider a unit circlecircle with radius 1 $C_0$ on the plane that does not enclose the origin. For $k\ge 1$, a circle $C_k$ is created by scaling and rotating $C_{k - 1}$ with respect to the origin. That is, both the radius and the distance to the origin are scaled by the same factor, and the centre of rotation is the... | <p>Consider a <strong class="tooltip">unit circle<span class="tooltiptext">circle with radius 1</span></strong> $C_0$ on the plane that does not enclose the origin. For $k\ge 1$, a circle $C_k$ is created by scaling and rotating $C_{k - 1}$ <b>with respect to the origin</b>. That is, both the radius and the distance to... | 0.7718678168 | Saturday, 8th June 2024, 02:00 pm | 332 | 35% | medium |
673 | Beds and Desks | At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom.
Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is either a sing... | At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom.
Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is either a sing... | <p>At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom.</p>
<p>Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is eit... | 700325380 | Sunday, 2nd June 2019, 07:00 am | 347 | 35% | medium |
318 | 2011 Nines | Consider the real number $\sqrt 2 + \sqrt 3$.
When we calculate the even powers of $\sqrt 2 + \sqrt 3$
we get:
$(\sqrt 2 + \sqrt 3)^2 = 9.898979485566356 \cdots $
$(\sqrt 2 + \sqrt 3)^4 = 97.98979485566356 \cdots $
$(\sqrt 2 + \sqrt 3)^6 = 969.998969071069263 \cdots $
$(\sqrt 2 + \sqrt 3)^8 = 9601.99989585502907 \cdots... | Consider the real number $\sqrt 2 + \sqrt 3$.
When we calculate the even powers of $\sqrt 2 + \sqrt 3$
we get:
$(\sqrt 2 + \sqrt 3)^2 = 9.898979485566356 \cdots $
$(\sqrt 2 + \sqrt 3)^4 = 97.98979485566356 \cdots $
$(\sqrt 2 + \sqrt 3)^6 = 969.998969071069263 \cdots $
$(\sqrt 2 + \sqrt 3)^8 = 9601.99989585502907 \cdots... | <p>
Consider the real number $\sqrt 2 + \sqrt 3$.<br/>
When we calculate the even powers of $\sqrt 2 + \sqrt 3$
we get:<br/>
$(\sqrt 2 + \sqrt 3)^2 = 9.898979485566356 \cdots $<br/>
$(\sqrt 2 + \sqrt 3)^4 = 97.98979485566356 \cdots $<br/>
$(\sqrt 2 + \sqrt 3)^6 = 969.998969071069263 \cdots $<br/>
$(\sqrt 2 + \sqrt 3)^8... | 709313889 | Saturday, 1st January 2011, 04:00 pm | 1051 | 50% | medium |
58 | Spiral Primes | Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 2643 44 45 46 47 48 49
It is interesting to note that the odd squares lie along ... | Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 2643 44 45 46 47 48 49
It is interesting to note that the odd squares lie along ... | <p>Starting with $1$ and spiralling anticlockwise in the following way, a square spiral with side length $7$ is formed.</p>
<p class="center monospace"><span class="red"><b>37</b></span> 36 35 34 33 32 <span class="red"><b>31</b></span><br/>
38 <span class="red"><b>17</b></span> 16 15 14 <span class="red"><b>13</b></sp... | 26241 | Friday, 5th December 2003, 06:00 pm | 44286 | 5% | easy |
792 | Too Many Twos | We define $\nu_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, $\nu_2(24) = 3$.
Define $\displaystyle S(n) = \sum_{k = 1}^n (-2)^k\binom{2k}k$ and $u(n) = \nu_2\Big(3S(n)+4\Big)$.
For example, when $n = 4$ then $S(4) = 980$ and $3S(4) + 4 = 2944 = 2^7 \cdot 23$, hence $u(4) = 7$.
You ... | We define $\nu_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, $\nu_2(24) = 3$.
Define $\displaystyle S(n) = \sum_{k = 1}^n (-2)^k\binom{2k}k$ and $u(n) = \nu_2\Big(3S(n)+4\Big)$.
For example, when $n = 4$ then $S(4) = 980$ and $3S(4) + 4 = 2944 = 2^7 \cdot 23$, hence $u(4) = 7$.
You ... | <p>
We define $\nu_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, $\nu_2(24) = 3$.
</p>
<p>
Define $\displaystyle S(n) = \sum_{k = 1}^n (-2)^k\binom{2k}k$ and $u(n) = \nu_2\Big(3S(n)+4\Big)$.
</p>
<p>
For example, when $n = 4$ then $S(4) = 980$ and $3S(4) + 4 = 2944 = 2^7 \cdot 23$, henc... | 2500500025183626 | Sunday, 3rd April 2022, 05:00 am | 157 | 100% | hard |
536 | Modulo Power Identity | Let $S(n)$ be the sum of all positive integers $m$ not exceeding $n$ having the following property:
$a^{m + 4} \equiv a \pmod m$ for all integers $a$.
The values of $m \le 100$ that satisfy this property are $1, 2, 3, 5$ and $21$, thus $S(100) = 1+2+3+5+21 = 32$.
You are given $S(10^6) = 22868117$.
Find $S(10^{12})... | Let $S(n)$ be the sum of all positive integers $m$ not exceeding $n$ having the following property:
$a^{m + 4} \equiv a \pmod m$ for all integers $a$.
The values of $m \le 100$ that satisfy this property are $1, 2, 3, 5$ and $21$, thus $S(100) = 1+2+3+5+21 = 32$.
You are given $S(10^6) = 22868117$.
Find $S(10^{12})... | <p>
Let $S(n)$ be the sum of all positive integers $m$ not exceeding $n$ having the following property:<br/>
$a^{m + 4} \equiv a \pmod m$ for all integers $a$.
</p>
<p>
The values of $m \le 100$ that satisfy this property are $1, 2, 3, 5$ and $21$, thus $S(100) = 1+2+3+5+21 = 32$.<br/>
You are given $S(10^6) = 22868117... | 3557005261906288 | Saturday, 28th November 2015, 07:00 pm | 327 | 60% | hard |
273 | Sum of Squares | Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
For $N=65$ there are two solutions:
$a=1$, $b=8$ and $a=4$, $b=7$.
We call $S(N)$ the sum of the values of $a$ of all solutions of $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
Thus $S(65) = 1 + 4 = 5$.
Find $\sum ... | Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
For $N=65$ there are two solutions:
$a=1$, $b=8$ and $a=4$, $b=7$.
We call $S(N)$ the sum of the values of $a$ of all solutions of $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
Thus $S(65) = 1 + 4 = 5$.
Find $\sum ... | <p>Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.</p>
<p>For $N=65$ there are two solutions:</p>
<p>$a=1$, $b=8$ and $a=4$, $b=7$.</p>
<p>We call $S(N)$ the sum of the values of $a$ of all solutions of $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.</p>
<p>Thus $... | 2032447591196869022 | Saturday, 9th January 2010, 01:00 pm | 1556 | 70% | hard |
16 | Power Digit Sum | $2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$.
What is the sum of the digits of the number $2^{1000}$? | $2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$.
What is the sum of the digits of the number $2^{1000}$? | <p>$2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$.</p>
<p>What is the sum of the digits of the number $2^{1000}$?</p> | 1366 | Friday, 3rd May 2002, 06:00 pm | 246503 | 5% | easy |
433 | Steps in Euclid's Algorithm | Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with Euclid's algorithm. More formally:$x_1 = y_0$, $y_1 = x_0 \bmod y_0$$x_n = y_{n-1}$, $y_n = x_{n-1} \bmod y_{n-1}$
$E(x_0, y_0)$ is the smallest $n$ such that $y_n = 0$.
We have $E(1,1) = 1$, $E(10,6) = ... | Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with Euclid's algorithm. More formally:$x_1 = y_0$, $y_1 = x_0 \bmod y_0$$x_n = y_{n-1}$, $y_n = x_{n-1} \bmod y_{n-1}$
$E(x_0, y_0)$ is the smallest $n$ such that $y_n = 0$.
We have $E(1,1) = 1$, $E(10,6) = ... | <p>
Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with <strong>Euclid's algorithm</strong>. More formally:<br/>$x_1 = y_0$, $y_1 = x_0 \bmod y_0$<br/>$x_n = y_{n-1}$, $y_n = x_{n-1} \bmod y_{n-1}$<br/>
$E(x_0, y_0)$ is the smallest $n$ such that $y_n = 0$.... | 326624372659664 | Saturday, 22nd June 2013, 04:00 pm | 503 | 65% | hard |
504 | Square on the Inside | Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:
$A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1 \le a, b, c, d \le m$ and $a, b, c, d, m$ are integers.
It can be shown that for $m = 4$ there are exactly $256$ valid ways to construct $ABCD$. Of these $256$ ... | Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:
$A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1 \le a, b, c, d \le m$ and $a, b, c, d, m$ are integers.
It can be shown that for $m = 4$ there are exactly $256$ valid ways to construct $ABCD$. Of these $256$ ... | <p>Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:</p>
<p>$A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1 \le a, b, c, d \le m$ and $a, b, c, d, m$ are integers.</p>
<p>It can be shown that for $m = 4$ there are exactly $256$ valid ways to construct $ABCD$... | 694687 | Saturday, 21st February 2015, 10:00 pm | 3406 | 15% | easy |
829 | Integral Fusion | Given any integer $n \gt 1$ a binary factor tree $T(n)$ is defined to be:
A tree with the single node $n$ when $n$ is prime.
A binary tree that has root node $n$, left subtree $T(a)$ and right subtree $T(b)$, when $n$ is not prime. Here $a$ and $b$ are positive integers such that $n = ab$, $a\le b$ and $b-a$ is the sm... | Given any integer $n \gt 1$ a binary factor tree $T(n)$ is defined to be:
A tree with the single node $n$ when $n$ is prime.
A binary tree that has root node $n$, left subtree $T(a)$ and right subtree $T(b)$, when $n$ is not prime. Here $a$ and $b$ are positive integers such that $n = ab$, $a\le b$ and $b-a$ is the sm... | <p>Given any integer $n \gt 1$ a <dfn>binary factor tree</dfn> $T(n)$ is defined to be:</p>
<ul>
<li>A tree with the single node $n$ when $n$ is prime.</li>
<li>A binary tree that has root node $n$, left subtree $T(a)$ and right subtree $T(b)$, when $n$ is not prime. Here $a$ and $b$ are positive integers such that $n ... | 41768797657018024 | Saturday, 11th February 2023, 07:00 pm | 220 | 45% | medium |
622 | Riffle Shuffles | A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the left hand and the bottom half taken in the right hand. Next, the cards are interleaved exactly, with the top card in the right half inserted just after the top card in the left half, the 2nd card in t... | A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the left hand and the bottom half taken in the right hand. Next, the cards are interleaved exactly, with the top card in the right half inserted just after the top card in the left half, the 2nd card in t... | <p>
A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the left hand and the bottom half taken in the right hand. Next, the cards are interleaved exactly, with the top card in the right half inserted just after the top card in the left half, the 2nd card ... | 3010983666182123972 | Sunday, 11th March 2018, 07:00 am | 1918 | 15% | easy |
138 | Special Isosceles Triangles | Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$.
By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2 - 8^2} = 15$, which is one less than the base length.
With $b = 272$ and $L = 305$, we get $h = 273$, which is one more than the base length,... | Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$.
By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2 - 8^2} = 15$, which is one less than the base length.
With $b = 272$ and $L = 305$, we get $h = 273$, which is one more than the base length,... | <p>Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$.</p>
<div class="center">
<img alt="" class="dark_img" height="228" src="resources/images/0138.png?1678992052" width="230"/></div>
<p>By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2 - 8^2} ... | 1118049290473932 | Saturday, 20th January 2007, 11:00 am | 6556 | 45% | medium |
703 | Circular Logic II | Given an integer $n$, $n \geq 3$, let $B=\{\mathrm{false},\mathrm{true}\}$ and let $B^n$ be the set of sequences of $n$ values from $B$. The function $f$ from $B^n$ to $B^n$ is defined by $f(b_1 \dots b_n) = c_1 \dots c_n$ where:
$c_i = b_{i+1}$ for $1 \leq i < n$.
$c_n = b_1 \;\mathrm{AND}\; (b_2 \;\mathrm{XOR}\; b_3)... | Given an integer $n$, $n \geq 3$, let $B=\{\mathrm{false},\mathrm{true}\}$ and let $B^n$ be the set of sequences of $n$ values from $B$. The function $f$ from $B^n$ to $B^n$ is defined by $f(b_1 \dots b_n) = c_1 \dots c_n$ where:
$c_i = b_{i+1}$ for $1 \leq i < n$.
$c_n = b_1 \;\mathrm{AND}\; (b_2 \;\mathrm{XOR}\; b_3)... | <p>Given an integer $n$, $n \geq 3$, let $B=\{\mathrm{false},\mathrm{true}\}$ and let $B^n$ be the set of sequences of $n$ values from $B$. The function $f$ from $B^n$ to $B^n$ is defined by $f(b_1 \dots b_n) = c_1 \dots c_n$ where:</p>
<ul><li>$c_i = b_{i+1}$ for $1 \leq i < n$.</li>
<li>$c_n = b_1 \;\mathrm{AND}\;... | 843437991 | Saturday, 22nd February 2020, 10:00 pm | 347 | 45% | medium |
288 | An Enormous Factorial | For any prime $p$ the number $N(p, q)$ is defined by
$N(p, q) = \sum_{n = 0}^q T_n \cdot p^n$
with $T_n$ generated by the following random number generator:
$S_0 = 290797$
$S_{n + 1} = S_n^2 \bmod 50515093$
$T_n = S_n \bmod p$
Let $\operatorname{Nfac}(p, q)$ be the factorial of $N(p, q)$.
Let $\operatorname{NF}(p, q... | For any prime $p$ the number $N(p, q)$ is defined by
$N(p, q) = \sum_{n = 0}^q T_n \cdot p^n$
with $T_n$ generated by the following random number generator:
$S_0 = 290797$
$S_{n + 1} = S_n^2 \bmod 50515093$
$T_n = S_n \bmod p$
Let $\operatorname{Nfac}(p, q)$ be the factorial of $N(p, q)$.
Let $\operatorname{NF}(p, q... | <p>
For any prime $p$ the number $N(p, q)$ is defined by
$N(p, q) = \sum_{n = 0}^q T_n \cdot p^n$<br/>
with $T_n$ generated by the following random number generator:</p>
<p>
$S_0 = 290797$<br/>
$S_{n + 1} = S_n^2 \bmod 50515093$<br/>
$T_n = S_n \bmod p$
</p>
<p>
Let $\operatorname{Nfac}(p, q)$ be the factorial of $N(p,... | 605857431263981935 | Saturday, 17th April 2010, 01:00 pm | 1859 | 35% | medium |
452 | Long Products | Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn't exceed $m$.
$F(10, 10) = 571$.
$F(10^6, 10^6) \bmod 1\,234\,567\,891 = 252903833$.
Find $F(10^9, 10^9) \bmod 1\,234\,567\,891$. | Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn't exceed $m$.
$F(10, 10) = 571$.
$F(10^6, 10^6) \bmod 1\,234\,567\,891 = 252903833$.
Find $F(10^9, 10^9) \bmod 1\,234\,567\,891$. | <p>Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn't exceed $m$.</p>
<p>$F(10, 10) = 571$.</p>
<p>$F(10^6, 10^6) \bmod 1\,234\,567\,891 = 252903833$.</p>
<p>Find $F(10^9, 10^9) \bmod 1\,234\,567\,891$.</p> | 345558983 | Saturday, 28th December 2013, 01:00 pm | 652 | 45% | medium |
529 | $10$-substrings | A $10$-substring of a number is a substring of its digits that sum to $10$. For example, the $10$-substrings of the number $3523014$ are:
3523014
3523014
3523014
3523014
A number is called $10$-substring-friendly if every one of its digits belongs to a $10$-substring. For example, $3523014$ is $10$-substring-friendly, ... | A $10$-substring of a number is a substring of its digits that sum to $10$. For example, the $10$-substrings of the number $3523014$ are:
3523014
3523014
3523014
3523014
A number is called $10$-substring-friendly if every one of its digits belongs to a $10$-substring. For example, $3523014$ is $10$-substring-friendly, ... | <p>A <dfn>$10$-substring</dfn> of a number is a substring of its digits that sum to $10$. For example, the $10$-substrings of the number $3523014$ are:</p>
<ul style="list-style-type:none;"><li><b><u>352</u></b>3014</li>
<li>3<b><u>523</u></b>014</li>
<li>3<b><u>5230</u></b>14</li>
<li>35<b><u>23014</u></b></li></ul>
<... | 23624465 | Saturday, 10th October 2015, 10:00 pm | 283 | 85% | hard |
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