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 |
|---|---|---|---|---|---|---|---|---|---|
569 | Prime Mountain Range | A mountain range consists of a line of mountains with slopes of exactly $45^\circ$, and heights governed by the prime numbers, $p_n$. The up-slope of the $k$th mountain is of height $p_{2k - 1}$, and the downslope is $p_{2k}$. The first few foot-hills of this range are illustrated below.
Tenzing sets out to climb ea... | A mountain range consists of a line of mountains with slopes of exactly $45^\circ$, and heights governed by the prime numbers, $p_n$. The up-slope of the $k$th mountain is of height $p_{2k - 1}$, and the downslope is $p_{2k}$. The first few foot-hills of this range are illustrated below.
Tenzing sets out to climb ea... | <p>A <dfn>mountain range</dfn> consists of a line of mountains with slopes of exactly $45^\circ$, and heights governed by the prime numbers, $p_n$. The up-slope of the $k$<sup>th</sup> mountain is of height $p_{2k - 1}$, and the downslope is $p_{2k}$. The first few foot-hills of this range are illustrated below.</p>
<d... | 21025060 | Saturday, 10th September 2016, 07:00 pm | 442 | 45% | medium |
461 | Almost Pi | Let $f_n(k) = e^{k/n} - 1$, for all non-negative integers $k$.
Remarkably, $f_{200}(6)+f_{200}(75)+f_{200}(89)+f_{200}(226)=\underline{3.1415926}44529\cdots\approx\pi$.
In fact, it is the best approximation of $\pi$ of the form $f_n(a) + f_n(b) + f_n(c) + f_n(d)$ for $n=200$.
Let $g(n)=a^2 + b^2 + c^2 + d^2$ for $a, b,... | Let $f_n(k) = e^{k/n} - 1$, for all non-negative integers $k$.
Remarkably, $f_{200}(6)+f_{200}(75)+f_{200}(89)+f_{200}(226)=\underline{3.1415926}44529\cdots\approx\pi$.
In fact, it is the best approximation of $\pi$ of the form $f_n(a) + f_n(b) + f_n(c) + f_n(d)$ for $n=200$.
Let $g(n)=a^2 + b^2 + c^2 + d^2$ for $a, b,... | <p>Let $f_n(k) = e^{k/n} - 1$, for all non-negative integers $k$.</p>
<p>Remarkably, $f_{200}(6)+f_{200}(75)+f_{200}(89)+f_{200}(226)=\underline{3.1415926}44529\cdots\approx\pi$.</p>
<p>In fact, it is the best approximation of $\pi$ of the form $f_n(a) + f_n(b) + f_n(c) + f_n(d)$ for $n=200$.</p>
<p>Let $g(n)=a^2 + b^2... | 159820276 | Saturday, 1st March 2014, 04:00 pm | 1350 | 30% | easy |
20 | Factorial Digit Sum | $n!$ means $n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$.
For example, $10! = 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 3628800$,and the sum of the digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.
Find the sum of the digits in the number $100!$. | $n!$ means $n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$.
For example, $10! = 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 3628800$,and the sum of the digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.
Find the sum of the digits in the number $100!$. | <p>$n!$ means $n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$.</p>
<p>For example, $10! = 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 3628800$,<br/>and the sum of the digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.</p>
<p>Find the sum of the digits in the number $100!$.</p> | 648 | Friday, 21st June 2002, 06:00 pm | 213193 | 5% | easy |
163 | Cross-hatched Triangles | Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size $1$ triangle in the sketch below.
Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using size $1$ triangles as ... | Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size $1$ triangle in the sketch below.
Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using size $1$ triangles as ... | <p>Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the <i>size $1$</i> triangle in the sketch below.</p>
<div class="center"><img alt="" class="dark_img" src="resources/images/0163.gif?1678992055"/></div>
<p>Sixteen triangles of either d... | 343047 | Saturday, 13th October 2007, 02:00 am | 2079 | 70% | hard |
17 | Number Letter Counts | If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.
If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, $3... | If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.
If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, $3... | <p>If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.</p>
<p>If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used? </p>
<br/><p class="note"><b>NOTE:</b> Do no... | 21124 | Friday, 17th May 2002, 06:00 pm | 164064 | 5% | easy |
824 | Chess Sliders | A Slider is a chess piece that can move one square left or right.
This problem uses a cylindrical chess board where the left hand edge of the board is connected to the right hand edge. This means that a Slider that is on the left hand edge of the chess board can move to the right hand edge of the same row and vice vers... | A Slider is a chess piece that can move one square left or right.
This problem uses a cylindrical chess board where the left hand edge of the board is connected to the right hand edge. This means that a Slider that is on the left hand edge of the chess board can move to the right hand edge of the same row and vice vers... | <p>A <dfn>Slider</dfn> is a chess piece that can move one square left or right.</p>
<p>This problem uses a cylindrical chess board where the left hand edge of the board is connected to the right hand edge. This means that a Slider that is on the left hand edge of the chess board can move to the right hand edge of the s... | 26532152736197 | Sunday, 8th January 2023, 04:00 am | 135 | 95% | hard |
309 | Integer Ladders | In the classic "Crossing Ladders" problem, we are given the lengths $x$ and $y$ of two ladders resting on the opposite walls of a narrow, level street. We are also given the height $h$ above the street where the two ladders cross and we are asked to find the width of the street ($w$).
Here, we are only concerned with ... | In the classic "Crossing Ladders" problem, we are given the lengths $x$ and $y$ of two ladders resting on the opposite walls of a narrow, level street. We are also given the height $h$ above the street where the two ladders cross and we are asked to find the width of the street ($w$).
Here, we are only concerned with ... | <p>In the classic "Crossing Ladders" problem, we are given the lengths $x$ and $y$ of two ladders resting on the opposite walls of a narrow, level street. We are also given the height $h$ above the street where the two ladders cross and we are asked to find the width of the street ($w$).</p>
<div align="center"><img al... | 210139 | Saturday, 6th November 2010, 04:00 pm | 915 | 50% | medium |
896 | Divisible Ranges | A contiguous range of positive integers is called a divisible range if all the integers in the range can be arranged in a row such that the $n$-th term is a multiple of $n$.
For example, the range $[6..9]$ is a divisible range because we can arrange the numbers as $7,6,9,8$.
In fact, it is the $4$th divisible range of ... | A contiguous range of positive integers is called a divisible range if all the integers in the range can be arranged in a row such that the $n$-th term is a multiple of $n$.
For example, the range $[6..9]$ is a divisible range because we can arrange the numbers as $7,6,9,8$.
In fact, it is the $4$th divisible range of ... | <p>
A contiguous range of positive integers is called a <dfn>divisible range</dfn> if all the integers in the range can be arranged in a row such that the $n$-th term is a multiple of $n$.<br/>
For example, the range $[6..9]$ is a divisible range because we can arrange the numbers as $7,6,9,8$.<br/>
In fact, it is the ... | 274229635640 | Saturday, 22nd June 2024, 08:00 pm | 217 | 50% | medium |
496 | Incenter and Circumcenter of Triangle | Given an integer sided triangle $ABC$:
Let $I$ be the incenter of $ABC$.
Let $D$ be the intersection between the line $AI$ and the circumcircle of $ABC$ ($A \ne D$).
We define $F(L)$ as the sum of $BC$ for the triangles $ABC$ that satisfy $AC = DI$ and $BC \le L$.
For example, $F(15) = 45$ because the triangles $ABC$ w... | Given an integer sided triangle $ABC$:
Let $I$ be the incenter of $ABC$.
Let $D$ be the intersection between the line $AI$ and the circumcircle of $ABC$ ($A \ne D$).
We define $F(L)$ as the sum of $BC$ for the triangles $ABC$ that satisfy $AC = DI$ and $BC \le L$.
For example, $F(15) = 45$ because the triangles $ABC$ w... | <p>Given an integer sided triangle $ABC$:<br/>
Let $I$ be the incenter of $ABC$.<br/>
Let $D$ be the intersection between the line $AI$ and the circumcircle of $ABC$ ($A \ne D$).</p>
<p>We define $F(L)$ as the sum of $BC$ for the triangles $ABC$ that satisfy $AC = DI$ and $BC \le L$.</p>
<p>For example, $F(15) = 45$ be... | 2042473533769142717 | Sunday, 4th January 2015, 01:00 am | 361 | 50% | medium |
326 | Modulo Summations | Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$.
So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$.
Let $f(N, M)$ represent the number of pairs $(p, q)$ such that:
$$
\def\htmltext#1{\style{font-family:inherit;}{\text{... | Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$.
So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$.
Let $f(N, M)$ represent the number of pairs $(p, q)$ such that:
$$
\def\htmltext#1{\style{font-family:inherit;}{\text{... | <p>
Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$.
</p>
<p>
So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$.
</p>
<p>Let $f(N, M)$ represent the number of pairs $(p, q)$ such that: </p>
<p>
$$
\def\htmltext#1{\style{fon... | 1966666166408794329 | Saturday, 26th February 2011, 04:00 pm | 593 | 55% | medium |
74 | Digit Factorial Chains | The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$:
$$1! + 4! + 5! = 1 + 24 + 120 = 145.$$
Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops that exist:
... | The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$:
$$1! + 4! + 5! = 1 + 24 + 120 = 145.$$
Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops that exist:
... | <p>The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$:
$$1! + 4! + 5! = 1 + 24 + 120 = 145.$$</p>
<p>Perhaps less well known is $169$, in that it produces the longest chain of numbers that link back to $169$; it turns out that there are only three such loops th... | 402 | Friday, 16th July 2004, 06:00 pm | 29660 | 15% | easy |
76 | Counting Summations | It is possible to write five as a sum in exactly six different ways:
\begin{align}
&4 + 1\\
&3 + 2\\
&3 + 1 + 1\\
&2 + 2 + 1\\
&2 + 1 + 1 + 1\\
&1 + 1 + 1 + 1 + 1
\end{align}
How many different ways can one hundred be written as a sum of at least two positive integers? | It is possible to write five as a sum in exactly six different ways:
\begin{align}
&4 + 1\\
&3 + 2\\
&3 + 1 + 1\\
&2 + 2 + 1\\
&2 + 1 + 1 + 1\\
&1 + 1 + 1 + 1 + 1
\end{align}
How many different ways can one hundred be written as a sum of at least two positive integers? | <p>It is possible to write five as a sum in exactly six different ways:</p>
\begin{align}
&4 + 1\\
&3 + 2\\
&3 + 1 + 1\\
&2 + 2 + 1\\
&2 + 1 + 1 + 1\\
&1 + 1 + 1 + 1 + 1
\end{align}
<p>How many different ways can one hundred be written as a sum of at least two positive integers?</p> | 190569291 | Friday, 13th August 2004, 06:00 pm | 31588 | 10% | easy |
516 | $5$-smooth Totients | $5$-smooth numbers are numbers whose largest prime factor doesn't exceed $5$.
$5$-smooth numbers are also called Hamming numbers.
Let $S(L)$ be the sum of the numbers $n$ not exceeding $L$ such that Euler's totient function $\phi(n)$ is a Hamming number.
$S(100)=3728$.
Find $S(10^{12})$. Give your answer modulo $2^{3... | $5$-smooth numbers are numbers whose largest prime factor doesn't exceed $5$.
$5$-smooth numbers are also called Hamming numbers.
Let $S(L)$ be the sum of the numbers $n$ not exceeding $L$ such that Euler's totient function $\phi(n)$ is a Hamming number.
$S(100)=3728$.
Find $S(10^{12})$. Give your answer modulo $2^{3... | <p>
$5$-smooth numbers are numbers whose largest prime factor doesn't exceed $5$.<br/>
$5$-smooth numbers are also called Hamming numbers.<br/>
Let $S(L)$ be the sum of the numbers $n$ not exceeding $L$ such that Euler's totient function $\phi(n)$ is a Hamming number.<br/>
$S(100)=3728$.
</p>
<p>
Find $S(10^{12})$. Giv... | 939087315 | Sunday, 17th May 2015, 10:00 am | 1715 | 20% | easy |
759 | A Squared Recurrence Relation | The function $f$ is defined for all positive integers as follows:
\begin{align*}
f(1) &= 1\\
f(2n) &= 2f(n)\\
f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n)
\end{align*}
It can be proven that $f(n)$ is integer for all values of $n$.
The function $S(n)$ is defined as $S(n) = \displaystyle \sum_{i=1}^n f(i) ^2$.
For example, $S... | The function $f$ is defined for all positive integers as follows:
\begin{align*}
f(1) &= 1\\
f(2n) &= 2f(n)\\
f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n)
\end{align*}
It can be proven that $f(n)$ is integer for all values of $n$.
The function $S(n)$ is defined as $S(n) = \displaystyle \sum_{i=1}^n f(i) ^2$.
For example, $S... | <p>The function $f$ is defined for all positive integers as follows:</p>
\begin{align*}
f(1) &= 1\\
f(2n) &= 2f(n)\\
f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n)
\end{align*}
<p>It can be proven that $f(n)$ is integer for all values of $n$.</p>
<p>The function $S(n)$ is defined as $S(n) = \displaystyle \sum_{i=1}^n f(i) ^2$... | 282771304 | Saturday, 12th June 2021, 11:00 pm | 599 | 25% | easy |
889 | Rational Blancmange | Recall the blancmange function from Problem 226: $T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$, where $s(x)$ is the distance from $x$ to the nearest integer.
For positive integers $k, t, r$, we write $$F(k, t, r) = (2^{2k} - 1)T\left(\frac{(2^t + 1)^r}{2^k + 1}\right).$$ It can be shown that $F(k, t, r)$ is ... | Recall the blancmange function from Problem 226: $T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$, where $s(x)$ is the distance from $x$ to the nearest integer.
For positive integers $k, t, r$, we write $$F(k, t, r) = (2^{2k} - 1)T\left(\frac{(2^t + 1)^r}{2^k + 1}\right).$$ It can be shown that $F(k, t, r)$ is ... | <p>
Recall the blancmange function from <a href="problem=226">Problem 226</a>: $T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$, where $s(x)$ is the distance from $x$ to the nearest integer.</p>
<p>
For positive integers $k, t, r$, we write $$F(k, t, r) = (2^{2k} - 1)T\left(\frac{(2^t + 1)^r}{2^k + 1}\right).$$ ... | 424315113 | Saturday, 4th May 2024, 11:00 pm | 125 | 70% | hard |
578 | Integers with Decreasing Prime Powers | Any positive integer can be written as a product of prime powers: $p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$,
where $p_i$ are distinct prime integers, $a_i \gt 0$ and $p_i \lt p_j$ if $i \lt j$.
A decreasing prime power positive integer is one for which $a_i \ge a_j$ if $i \lt j$.
For example, $1$, $2$... | Any positive integer can be written as a product of prime powers: $p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$,
where $p_i$ are distinct prime integers, $a_i \gt 0$ and $p_i \lt p_j$ if $i \lt j$.
A decreasing prime power positive integer is one for which $a_i \ge a_j$ if $i \lt j$.
For example, $1$, $2$... | <p>Any positive integer can be written as a product of prime powers: $p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$,<br/>
where $p_i$ are distinct prime integers, $a_i \gt 0$ and $p_i \lt p_j$ if $i \lt j$.</p>
<p>A <dfn>decreasing prime power</dfn> positive integer is one for which $a_i \ge a_j$ if $i \lt... | 9219696799346 | Saturday, 19th November 2016, 10:00 pm | 267 | 80% | hard |
861 | Products of Bi-Unitary Divisors | A unitary divisor of a positive integer $n$ is a divisor $d$ of $n$ such that $\gcd\left(d,\frac{n}{d}\right)=1$.
A bi-unitary divisor of $n$ is a divisor $d$ for which $1$ is the only unitary divisor of $d$ that is also a unitary divisor of $\frac{n}{d}$.
For example, $2$ is a bi-unitary divisor of $8$, because the un... | A unitary divisor of a positive integer $n$ is a divisor $d$ of $n$ such that $\gcd\left(d,\frac{n}{d}\right)=1$.
A bi-unitary divisor of $n$ is a divisor $d$ for which $1$ is the only unitary divisor of $d$ that is also a unitary divisor of $\frac{n}{d}$.
For example, $2$ is a bi-unitary divisor of $8$, because the un... | <p>A <i>unitary divisor</i> of a positive integer $n$ is a divisor $d$ of $n$ such that $\gcd\left(d,\frac{n}{d}\right)=1$.</p>
<p>A <i>bi-unitary divisor</i> of $n$ is a divisor $d$ for which $1$ is the only unitary divisor of $d$ that is also a unitary divisor of $\frac{n}{d}$.</p>
<p>For example, $2$ is a bi-unitary... | 672623540591 | Saturday, 28th October 2023, 02:00 pm | 217 | 40% | medium |
637 | Flexible Digit Sum | Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$, and then carrying out the additions.
For example, from $n=123_{10}$ ($n$ in base $10$) we can construct the four base $10$ integers $123_{10}$, $1+23=24_{10}$, $12... | Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$, and then carrying out the additions.
For example, from $n=123_{10}$ ($n$ in base $10$) we can construct the four base $10$ integers $123_{10}$, $1+23=24_{10}$, $12... | <p>
Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$, and then carrying out the additions.
</p>
<p>
For example, from $n=123_{10}$ ($n$ in base $10$) we can construct the four base $10$ integers $123_{10}$, $1+23=24... | 49000634845039 | Sunday, 23rd September 2018, 01:00 am | 354 | 45% | medium |
284 | Steady Squares | The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 3762 = 141376. Let's call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the... | The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 3762 = 141376. Let's call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the... | <p>The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 376<sup>2</sup> = 141376. Let's call a number with this property a steady square.</p>
<p>Steady squares can also be observed in other numbering systems. In the base 14 ... | 5a411d7b | Saturday, 27th March 2010, 01:00 am | 1397 | 55% | medium |
575 | Wandering Robots | It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the purpos... | It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the purpos... | <p>It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the pur... | 0.000989640561 | Saturday, 22nd October 2016, 01:00 pm | 645 | 35% | medium |
457 | A Polynomial Modulo the Square of a Prime | Let $f(n) = n^2 - 3n - 1$.
Let $p$ be a prime.
Let $R(p)$ be the smallest positive integer $n$ such that $f(n) \bmod p^2 = 0$ if such an integer $n$ exists, otherwise $R(p) = 0$.
Let $SR(L)$ be $\sum R(p)$ for all primes not exceeding $L$.
Find $SR(10^7)$. | Let $f(n) = n^2 - 3n - 1$.
Let $p$ be a prime.
Let $R(p)$ be the smallest positive integer $n$ such that $f(n) \bmod p^2 = 0$ if such an integer $n$ exists, otherwise $R(p) = 0$.
Let $SR(L)$ be $\sum R(p)$ for all primes not exceeding $L$.
Find $SR(10^7)$. | <p>
Let $f(n) = n^2 - 3n - 1$.<br/>
Let $p$ be a prime.<br/>
Let $R(p)$ be the smallest positive integer $n$ such that $f(n) \bmod p^2 = 0$ if such an integer $n$ exists, otherwise $R(p) = 0$.
</p>
<p>
Let $SR(L)$ be $\sum R(p)$ for all primes not exceeding $L$.
</p>
<p>
Find $SR(10^7)$.
</p> | 2647787126797397063 | Sunday, 2nd February 2014, 04:00 am | 903 | 35% | medium |
747 | Triangular Pizza | Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into $n$ pieces. She first chooses a point $P$ in the interior (not boundary) of the triangle pizza, and then performs $n$ cuts, which all start from $P$ and extend straight to the boundary of the pizza so that the $n$ pieces are all triangles and all... | Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into $n$ pieces. She first chooses a point $P$ in the interior (not boundary) of the triangle pizza, and then performs $n$ cuts, which all start from $P$ and extend straight to the boundary of the pizza so that the $n$ pieces are all triangles and all... | <p>Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into $n$ pieces. She first chooses a point $P$ in the interior (not boundary) of the triangle pizza, and then performs $n$ cuts, which all start from $P$ and extend straight to the boundary of the pizza so that the $n$ pieces are all triangles and ... | 681813395 | Sunday, 14th February 2021, 10:00 am | 210 | 60% | hard |
467 | Superinteger | An integer $s$ is called a superinteger of another integer $n$ if the digits of $n$ form a subsequenceA subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. of the digits of $s$.
For example, $2718281828$ is a superinteger of... | An integer $s$ is called a superinteger of another integer $n$ if the digits of $n$ form a subsequenceA subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. of the digits of $s$.
For example, $2718281828$ is a superinteger of... | <p>An integer $s$ is called a <dfn>superinteger</dfn> of another integer $n$ if the digits of $n$ form a <strong class="tooltip">subsequence<span class="tooltiptext">A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.</spa... | 775181359 | Sunday, 13th April 2014, 10:00 am | 491 | 50% | medium |
892 | Zebra Circles | Consider a circle where $2n$ distinct points have been marked on its circumference.
A cutting $C$ consists of connecting the $2n$ points with $n$ line segments, so that no two line segments intersect, including on their end points. The $n$ line segments then cut the circle into $n + 1$ pieces.
Each piece is painted ei... | Consider a circle where $2n$ distinct points have been marked on its circumference.
A cutting $C$ consists of connecting the $2n$ points with $n$ line segments, so that no two line segments intersect, including on their end points. The $n$ line segments then cut the circle into $n + 1$ pieces.
Each piece is painted ei... | <p>
Consider a circle where $2n$ distinct points have been marked on its circumference.</p>
<p>
A <i>cutting</i> $C$ consists of connecting the $2n$ points with $n$ line segments, so that no two line segments intersect, including on their end points. The $n$ line segments then cut the circle into $n + 1$ pieces.
Each p... | 469137427 | Sunday, 26th May 2024, 08:00 am | 154 | 60% | hard |
662 | Fibonacci Paths | Alice walks on a lattice grid. She can step from one lattice point $A (a,b)$ to another $B (a+x,b+y)$ providing distance $AB = \sqrt{x^2+y^2}$ is a Fibonacci number $\{1,2,3,5,8,13,\ldots\}$ and $x\ge 0,$ $y\ge 0$.
In the lattice grid below Alice can step from the blue point to any of the red points.
Let $F(W,H)$... | Alice walks on a lattice grid. She can step from one lattice point $A (a,b)$ to another $B (a+x,b+y)$ providing distance $AB = \sqrt{x^2+y^2}$ is a Fibonacci number $\{1,2,3,5,8,13,\ldots\}$ and $x\ge 0,$ $y\ge 0$.
In the lattice grid below Alice can step from the blue point to any of the red points.
Let $F(W,H)$... | <p>
Alice walks on a lattice grid. She can step from one lattice point $A (a,b)$ to another $B (a+x,b+y)$ providing distance $AB = \sqrt{x^2+y^2}$ is a Fibonacci number $\{1,2,3,5,8,13,\ldots\}$ and $x\ge 0,$ $y\ge 0$.
</p>
<p>
In the lattice grid below Alice can step from the blue point to any of the red points.<br/... | 860873428 | Sunday, 24th March 2019, 01:00 am | 895 | 25% | easy |
28 | Number Spiral Diagonals | Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:
21 22 23 24 25
20 7 8 9 10
19 6 1 2 11
18 5 4 3 1217 16 15 14 13
It can be verified that the sum of the numbers on the diagonals is $101$.
What is the sum of the numbers on the diagonals in a... | Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:
21 22 23 24 25
20 7 8 9 10
19 6 1 2 11
18 5 4 3 1217 16 15 14 13
It can be verified that the sum of the numbers on the diagonals is $101$.
What is the sum of the numbers on the diagonals in a... | <p>Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:</p>
<p class="monospace center"><span class="red"><b>21</b></span> 22 23 24 <span class="red"><b>25</b></span><br/>
20 <span class="red"><b>7</b></span> 8 <span class="red"><b>9</b></span> 10<br... | 669171001 | Friday, 11th October 2002, 06:00 pm | 116784 | 5% | easy |
250 | $250250$ | Find the number of non-empty subsets of $\{1^1, 2^2, 3^3,\dots, 250250^{250250}\}$, the sum of whose elements is divisible by $250$. Enter the rightmost $16$ digits as your answer. | Find the number of non-empty subsets of $\{1^1, 2^2, 3^3,\dots, 250250^{250250}\}$, the sum of whose elements is divisible by $250$. Enter the rightmost $16$ digits as your answer. | <p>Find the number of non-empty subsets of $\{1^1, 2^2, 3^3,\dots, 250250^{250250}\}$, the sum of whose elements is divisible by $250$. Enter the rightmost $16$ digits as your answer.</p> | 1425480602091519 | Saturday, 13th June 2009, 05:00 am | 3325 | 55% | medium |
293 | Pseudo-Fortunate Numbers | An even positive integer $N$ will be called admissible, if it is a power of $2$ or its distinct prime factors are consecutive primes.
The first twelve admissible numbers are $2,4,6,8,12,16,18,24,30,32,36,48$.
If $N$ is admissible, the smallest integer $M \gt 1$ such that $N+M$ is prime, will be called the pseudo-Fort... | An even positive integer $N$ will be called admissible, if it is a power of $2$ or its distinct prime factors are consecutive primes.
The first twelve admissible numbers are $2,4,6,8,12,16,18,24,30,32,36,48$.
If $N$ is admissible, the smallest integer $M \gt 1$ such that $N+M$ is prime, will be called the pseudo-Fort... | <p>
An even positive integer $N$ will be called admissible, if it is a power of $2$ or its distinct prime factors are consecutive primes.<br/>
The first twelve admissible numbers are $2,4,6,8,12,16,18,24,30,32,36,48$.
</p>
<p>
If $N$ is admissible, the smallest integer $M \gt 1$ such that $N+M$ is prime, will be called... | 2209 | Saturday, 22nd May 2010, 05:00 am | 3264 | 30% | easy |
478 | Mixtures | Let us consider mixtures of three substances: A, B and C. A mixture can be described by a ratio of the amounts of A, B, and C in it, i.e., $(a : b : c)$. For example, a mixture described by the ratio $(2 : 3 : 5)$ contains $20\%$ A, $30\%$ B and $50\%$ C.
For the purposes of this problem, we cannot separate the individ... | Let us consider mixtures of three substances: A, B and C. A mixture can be described by a ratio of the amounts of A, B, and C in it, i.e., $(a : b : c)$. For example, a mixture described by the ratio $(2 : 3 : 5)$ contains $20\%$ A, $30\%$ B and $50\%$ C.
For the purposes of this problem, we cannot separate the individ... | <p>Let us consider <b>mixtures</b> of three substances: <b>A</b>, <b>B</b> and <b>C</b>. A mixture can be described by a ratio of the amounts of <b>A</b>, <b>B</b>, and <b>C</b> in it, i.e., $(a : b : c)$. For example, a mixture described by the ratio $(2 : 3 : 5)$ contains $20\%$ <b>A</b>, $30\%$ <b>B</b> and $50\%$ <... | 59510340 | Saturday, 30th August 2014, 07:00 pm | 220 | 100% | hard |
266 | Pseudo Square Root | The divisors of $12$ are: $1,2,3,4,6$ and $12$.
The largest divisor of $12$ that does not exceed the square root of $12$ is $3$.
We shall call the largest divisor of an integer $n$ that does not exceed the square root of $n$ the pseudo square root ($\operatorname{PSR}$) of $n$.
It can be seen that $\operatorname{PSR}(3... | The divisors of $12$ are: $1,2,3,4,6$ and $12$.
The largest divisor of $12$ that does not exceed the square root of $12$ is $3$.
We shall call the largest divisor of an integer $n$ that does not exceed the square root of $n$ the pseudo square root ($\operatorname{PSR}$) of $n$.
It can be seen that $\operatorname{PSR}(3... | <p>
The divisors of $12$ are: $1,2,3,4,6$ and $12$.<br/>
The largest divisor of $12$ that does not exceed the square root of $12$ is $3$.<br/>
We shall call the largest divisor of an integer $n$ that does not exceed the square root of $n$ the pseudo square root ($\operatorname{PSR}$) of $n$.<br/>
It can be seen that $\... | 1096883702440585 | Saturday, 28th November 2009, 01:00 pm | 1891 | 65% | hard |
748 | Upside Down Diophantine Equation | Upside Down is a modification of the famous Pythagorean equation:
\begin{align}
\frac{1}{x^2}+\frac{1}{y^2}=\frac{13}{z^2}.
\end{align}
A solution $(x,y,z)$ to this equation with $x,y$ and $z$ positive integers is a primitive solution if $\gcd(x,y,z)=1$.
Let $S(N)$ be the sum of $x+y+z$ over primitive Upside Down ... | Upside Down is a modification of the famous Pythagorean equation:
\begin{align}
\frac{1}{x^2}+\frac{1}{y^2}=\frac{13}{z^2}.
\end{align}
A solution $(x,y,z)$ to this equation with $x,y$ and $z$ positive integers is a primitive solution if $\gcd(x,y,z)=1$.
Let $S(N)$ be the sum of $x+y+z$ over primitive Upside Down ... | <p>
Upside Down is a modification of the famous Pythagorean equation:
\begin{align}
\frac{1}{x^2}+\frac{1}{y^2}=\frac{13}{z^2}.
\end{align}
</p>
<p>
A solution $(x,y,z)$ to this equation with $x,y$ and $z$ positive integers is a primitive solution if $\gcd(x,y,z)=1$.
</p>
<p>
Let $S(N)$ be the sum of $x+y+z$ over prim... | 276402862 | Saturday, 20th February 2021, 01:00 pm | 327 | 40% | medium |
248 | Euler's Totient Function Equals 13! | The first number $n$ for which $\phi(n)=13!$ is $6227180929$.
Find the $150\,000$th such number. | The first number $n$ for which $\phi(n)=13!$ is $6227180929$.
Find the $150\,000$th such number. | <p>The first number $n$ for which $\phi(n)=13!$ is $6227180929$.</p>
<p>Find the $150\,000$<sup>th</sup> such number.</p> | 23507044290 | Saturday, 6th June 2009, 01:00 am | 1470 | 70% | hard |
667 | Moving Pentagon | After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same company. Most important for him is that the table can be pushed through his L-shaped corridor into the living room without having to be lifted from its table legs.
Unfortunately, the simple square model o... | After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same company. Most important for him is that the table can be pushed through his L-shaped corridor into the living room without having to be lifted from its table legs.
Unfortunately, the simple square model o... | <p>
After buying a <i>Gerver Sofa</i> from the <i>Moving Sofa Company</i>, Jack wants to buy a matching cocktail table from the same company. Most important for him is that the table can be pushed through his L-shaped corridor into the living room without having to be lifted from its table legs. <br>
Unfortunately, the... | 1.5276527928 | Saturday, 27th April 2019, 04:00 pm | 222 | 80% | hard |
229 | Four Representations Using Squares | Consider the number $3600$. It is very special, because
\begin{alignat}{2}
3600 &= 48^2 + &&36^2\\
3600 &= 20^2 + 2 \times &&40^2\\
3600 &= 30^2 + 3 \times &&30^2\\
3600 &= 45^2 + 7 \times &&15^2
\end{alignat}
Similarly, we find that $88201 = 99^2 + 280^2 = 287^2 + 2 \times 54^2 = 283^2 + 3 \times 52^2 = 197^2 + 7 \ti... | Consider the number $3600$. It is very special, because
\begin{alignat}{2}
3600 &= 48^2 + &&36^2\\
3600 &= 20^2 + 2 \times &&40^2\\
3600 &= 30^2 + 3 \times &&30^2\\
3600 &= 45^2 + 7 \times &&15^2
\end{alignat}
Similarly, we find that $88201 = 99^2 + 280^2 = 287^2 + 2 \times 54^2 = 283^2 + 3 \times 52^2 = 197^2 + 7 \ti... | <p>Consider the number $3600$. It is very special, because</p>
\begin{alignat}{2}
3600 &= 48^2 + &&36^2\\
3600 &= 20^2 + 2 \times &&40^2\\
3600 &= 30^2 + 3 \times &&30^2\\
3600 &= 45^2 + 7 \times &&15^2
\end{alignat}
<p>Similarly, we find that $88201 = 99^2 + 280^2 = 287^2 + 2 \times 54^2 = 283^2 + 3 \times 52^2 = 197... | 11325263 | Saturday, 24th January 2009, 09:00 am | 1628 | 70% | hard |
631 | Constrained Permutations | Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, ..., k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$; note that the empty permutation $()$ has length zero.
Define an occurrence of a permutation $p=(p_1 p_2 \cdots p_k)$ in a permutation $P=(P_1 P_2 \cdots P_n)$ to be a sequ... | Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, ..., k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$; note that the empty permutation $()$ has length zero.
Define an occurrence of a permutation $p=(p_1 p_2 \cdots p_k)$ in a permutation $P=(P_1 P_2 \cdots P_n)$ to be a sequ... | <p>Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, ..., k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$; note that the empty permutation $()$ has length zero.</p>
<p>Define an <dfn>occurrence</dfn> of a permutation $p=(p_1 p_2 \cdots p_k)$ in a permutation $P=(P_1 P_2 \cdo... | 869588692 | Sunday, 15th July 2018, 10:00 am | 208 | 65% | hard |
802 | Iterated Composition | Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots\ $.
Consider the function $f$ from $\Bbb R^2$ to $\Bbb R^2$ defined by $f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$, and its $n$-th iterated composition $f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$. For example $f^{(3)}(x, y) = f(f(f(x,... | Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots\ $.
Consider the function $f$ from $\Bbb R^2$ to $\Bbb R^2$ defined by $f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$, and its $n$-th iterated composition $f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$. For example $f^{(3)}(x, y) = f(f(f(x,... | <p>Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots\ $.</p>
<p>Consider the function $f$ from $\Bbb R^2$ to $\Bbb R^2$ defined by $f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$, and its $n$-th iterated composition $f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$. For example $f^{(3)}(x, y) ... | 973873727 | Sunday, 12th June 2022, 11:00 am | 278 | 35% | medium |
295 | Lenticular Holes | We call the convex area enclosed by two circles a lenticular hole if:
The centres of both circles are on lattice points.
The two circles intersect at two distinct lattice points.
The interior of the convex area enclosed by both circles does not contain any lattice points.
Consider the circles:
$C_0$: $x^2 + y^2 = 25$
... | We call the convex area enclosed by two circles a lenticular hole if:
The centres of both circles are on lattice points.
The two circles intersect at two distinct lattice points.
The interior of the convex area enclosed by both circles does not contain any lattice points.
Consider the circles:
$C_0$: $x^2 + y^2 = 25$
... | <p>We call the convex area enclosed by two circles a <dfn>lenticular hole</dfn> if:
</p><ul><li>The centres of both circles are on lattice points.</li>
<li>The two circles intersect at two distinct lattice points.</li>
<li>The interior of the convex area enclosed by both circles does not contain any lattice points.
</l... | 4884650818 | Saturday, 5th June 2010, 01:00 pm | 512 | 75% | hard |
23 | Non-Abundant Sums | A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.
A number $n$ is called deficient if the sum of its proper divisors is less than $n$ and... | A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.
A number $n$ is called deficient if the sum of its proper divisors is less than $n$ and... | <p>A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.</p>
<p>A number $n$ is called deficient if the sum of its proper divisors is less th... | 4179871 | Friday, 2nd August 2002, 06:00 pm | 113788 | 5% | easy |
211 | Divisor Square Sum | For a positive integer $n$, let $\sigma_2(n)$ be the sum of the squares of its divisors. For example,
$$\sigma_2(10) = 1 + 4 + 25 + 100 = 130.$$
Find the sum of all $n$, $0 \lt n \lt 64\,000\,000$ such that $\sigma_2(n)$ is a perfect square. | For a positive integer $n$, let $\sigma_2(n)$ be the sum of the squares of its divisors. For example,
$$\sigma_2(10) = 1 + 4 + 25 + 100 = 130.$$
Find the sum of all $n$, $0 \lt n \lt 64\,000\,000$ such that $\sigma_2(n)$ is a perfect square. | <p>For a positive integer $n$, let $\sigma_2(n)$ be the sum of the squares of its divisors. For example,
$$\sigma_2(10) = 1 + 4 + 25 + 100 = 130.$$</p>
<p>Find the sum of all $n$, $0 \lt n \lt 64\,000\,000$ such that $\sigma_2(n)$ is a perfect square.</p> | 1922364685 | Saturday, 4th October 2008, 02:00 am | 4638 | 50% | medium |
90 | Cube Digit Pairs | Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.
For example, the square number $64$ could be formed:
In fact, by carefully choosing the digits o... | Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.
For example, the square number $64$ could be formed:
In fact, by carefully choosing the digits o... | <p>Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.</p>
<p>For example, the square number $64$ could be formed:</p>
<div class="center">
<img alt="... | 1217 | Friday, 4th March 2005, 06:00 pm | 12879 | 40% | medium |
149 | Maximum-sum Subsequence | Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is $16$ ($= 8 + 7 + 1$).
$-2$$5$$3$$2$$9$$-6$$5$$1$$3$$2$$7$$3$$-1$$8$$-4$$8$
Now, let us repeat the search, but on a much larger scale:
First, generate ... | Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is $16$ ($= 8 + 7 + 1$).
$-2$$5$$3$$2$$9$$-6$$5$$1$$3$$2$$7$$3$$-1$$8$$-4$$8$
Now, let us repeat the search, but on a much larger scale:
First, generate ... | <p>Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) <span style="white-space:nowrap;">is $16$ ($= 8 + 7 + 1$).</span></p>
<div class="center">
<table border="1" cellpadding="6" cellspacing="0" style="mar... | 52852124 | Friday, 13th April 2007, 10:00 pm | 5408 | 50% | medium |
486 | Palindrome-containing Strings | Let $F_5(n)$ be the number of strings $s$ such that:
$s$ consists only of '0's and '1's,
$s$ has length at most $n$, and
$s$ contains a palindromic substring of length at least $5$.
For example, $F_5(4) = 0$, $F_5(5) = 8$,
$F_5(6) = 42$ and $F_5(11) = 3844$.
Let $D(L)$ be the number of integers $n$ such that $5 \le n ... | Let $F_5(n)$ be the number of strings $s$ such that:
$s$ consists only of '0's and '1's,
$s$ has length at most $n$, and
$s$ contains a palindromic substring of length at least $5$.
For example, $F_5(4) = 0$, $F_5(5) = 8$,
$F_5(6) = 42$ and $F_5(11) = 3844$.
Let $D(L)$ be the number of integers $n$ such that $5 \le n ... | <p>Let $F_5(n)$ be the number of strings $s$ such that:</p>
<ul><li>$s$ consists only of '0's and '1's,
</li><li>$s$ has length at most $n$, and
</li><li>$s$ contains a palindromic substring of length at least $5$.
</li></ul><p>For example, $F_5(4) = 0$, $F_5(5) = 8$,
$F_5(6) = 42$ and $F_5(11) = 3844$.</p>
<p>Let $D(... | 11408450515 | Saturday, 25th October 2014, 07:00 pm | 291 | 70% | hard |
197 | A Recursively Defined Sequence | Given is the function $f(x) = \lfloor 2^{30.403243784 - x^2}\rfloor \times 10^{-9}$ ($\lfloor \, \rfloor$ is the floor-function),
the sequence $u_n$ is defined by $u_0 = -1$ and $u_{n + 1} = f(u_n)$.
Find $u_n + u_{n + 1}$ for $n = 10^{12}$.
Give your answer with $9$ digits after the decimal point. | Given is the function $f(x) = \lfloor 2^{30.403243784 - x^2}\rfloor \times 10^{-9}$ ($\lfloor \, \rfloor$ is the floor-function),
the sequence $u_n$ is defined by $u_0 = -1$ and $u_{n + 1} = f(u_n)$.
Find $u_n + u_{n + 1}$ for $n = 10^{12}$.
Give your answer with $9$ digits after the decimal point. | <p>Given is the function $f(x) = \lfloor 2^{30.403243784 - x^2}\rfloor \times 10^{-9}$ ($\lfloor \, \rfloor$ is the floor-function),<br/>
the sequence $u_n$ is defined by $u_0 = -1$ and $u_{n + 1} = f(u_n)$.</p>
<p>Find $u_n + u_{n + 1}$ for $n = 10^{12}$.<br/>
Give your answer with $9$ digits after the decimal point.<... | 1.710637717 | Friday, 6th June 2008, 10:00 pm | 5402 | 45% | medium |
135 | Same Differences | Given the positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression, the least value of the positive integer, $n$, for which the equation, $x^2 - y^2 - z^2 = n$, has exactly two solutions is $n = 27$:
$$34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27.$$
It turns out that $n = 1155$ is the leas... | Given the positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression, the least value of the positive integer, $n$, for which the equation, $x^2 - y^2 - z^2 = n$, has exactly two solutions is $n = 27$:
$$34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27.$$
It turns out that $n = 1155$ is the leas... | <p>Given the positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression, the least value of the positive integer, $n$, for which the equation, $x^2 - y^2 - z^2 = n$, has exactly two solutions is $n = 27$:
$$34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27.$$</p>
<p>It turns out that $n = 1155$ i... | 4989 | Friday, 29th December 2006, 06:00 pm | 7211 | 45% | medium |
525 | Rolling Ellipse | An ellipse $E(a, b)$ is given at its initial position by equation:
$\frac {x^2} {a^2} + \frac {(y - b)^2} {b^2} = 1$
The ellipse rolls without slipping along the $x$ axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis:
$F(a,b) = 2 \pi \... | An ellipse $E(a, b)$ is given at its initial position by equation:
$\frac {x^2} {a^2} + \frac {(y - b)^2} {b^2} = 1$
The ellipse rolls without slipping along the $x$ axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis:
$F(a,b) = 2 \pi \... | <p>An ellipse $E(a, b)$ is given at its initial position by equation:<br/>
$\frac {x^2} {a^2} + \frac {(y - b)^2} {b^2} = 1$</p>
<p>The ellipse rolls without slipping along the $x$ axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis:<br/... | 44.69921807 | Sunday, 13th September 2015, 10:00 am | 547 | 45% | medium |
646 | Bounded Divisors | Let $n$ be a natural number and $p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ its prime factorisation.
Define the Liouville function $\lambda(n)$ as $\lambda(n) = (-1)^{\sum\limits_{i=1}^{k}\alpha_i}$.
(i.e. $-1$ if the sum of the exponents $\alpha_i$ is odd and $1$ if the sum of the exponents is even. )
L... | Let $n$ be a natural number and $p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ its prime factorisation.
Define the Liouville function $\lambda(n)$ as $\lambda(n) = (-1)^{\sum\limits_{i=1}^{k}\alpha_i}$.
(i.e. $-1$ if the sum of the exponents $\alpha_i$ is odd and $1$ if the sum of the exponents is even. )
L... | <p>
Let $n$ be a natural number and $p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ its prime factorisation.<br/>
Define the <b>Liouville function</b> $\lambda(n)$ as $\lambda(n) = (-1)^{\sum\limits_{i=1}^{k}\alpha_i}$.<br/>
(i.e. $-1$ if the sum of the exponents $\alpha_i$ is odd and $1$ if the sum of the e... | 845218467 | Sunday, 9th December 2018, 04:00 am | 313 | 40% | medium |
68 | Magic 5-gon Ring | Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described ... | Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described ... | <p>Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0068_1.png?1678992052"/><br/></div>
<p>Working <b>clockwise</b>, and starting from the group of three with the numerically lowest extern... | 6531031914842725 | Friday, 23rd April 2004, 06:00 pm | 23263 | 25% | easy |
495 | Writing $n$ as the Product of $k$ Distinct Positive Integers | Let $W(n,k)$ be the number of ways in which $n$ can be written as the product of $k$ distinct positive integers.
For example, $W(144,4) = 7$. There are $7$ ways in which $144$ can be written as a product of $4$ distinct positive integers:
$144 = 1 \times 2 \times 4 \times 18$
$144 = 1 \times 2 \times 8 \times 9$
$144 =... | Let $W(n,k)$ be the number of ways in which $n$ can be written as the product of $k$ distinct positive integers.
For example, $W(144,4) = 7$. There are $7$ ways in which $144$ can be written as a product of $4$ distinct positive integers:
$144 = 1 \times 2 \times 4 \times 18$
$144 = 1 \times 2 \times 8 \times 9$
$144 =... | <p>Let $W(n,k)$ be the number of ways in which $n$ can be written as the product of $k$ distinct positive integers.</p>
<p>For example, $W(144,4) = 7$. There are $7$ ways in which $144$ can be written as a product of $4$ distinct positive integers:</p>
<p></p><ul><li>$144 = 1 \times 2 \times 4 \times 18$</li>
<li>$144 ... | 789107601 | Saturday, 27th December 2014, 10:00 pm | 357 | 100% | hard |
788 | Dominating Numbers | A dominating number is a positive integer that has more than half of its digits equal.
For example, $2022$ is a dominating number because three of its four digits are equal to $2$. But $2021$ is not a dominating number.
Let $D(N)$ be how many dominating numbers are less than $10^N$.
For example, $D(4) = 603$ and $D... | A dominating number is a positive integer that has more than half of its digits equal.
For example, $2022$ is a dominating number because three of its four digits are equal to $2$. But $2021$ is not a dominating number.
Let $D(N)$ be how many dominating numbers are less than $10^N$.
For example, $D(4) = 603$ and $D... | <p>
A <dfn>dominating number</dfn> is a positive integer that has more than half of its digits equal.
</p>
<p>
For example, $2022$ is a dominating number because three of its four digits are equal to $2$. But $2021$ is not a dominating number.
</p>
<p>
Let $D(N)$ be how many dominating numbers are less than $10^N$.
For... | 471745499 | Saturday, 5th March 2022, 04:00 pm | 1298 | 10% | easy |
498 | Remainder of Polynomial Division | For positive integers $n$ and $m$, we define two polynomials $F_n(x) = x^n$ and $G_m(x) = (x-1)^m$.
We also define a polynomial $R_{n,m}(x)$ as the remainder of the division of $F_n(x)$ by $G_m(x)$.
For example, $R_{6,3}(x) = 15x^2 - 24x + 10$.
Let $C(n, m, d)$ be the absolute value of the coefficient of the $d$-th deg... | For positive integers $n$ and $m$, we define two polynomials $F_n(x) = x^n$ and $G_m(x) = (x-1)^m$.
We also define a polynomial $R_{n,m}(x)$ as the remainder of the division of $F_n(x)$ by $G_m(x)$.
For example, $R_{6,3}(x) = 15x^2 - 24x + 10$.
Let $C(n, m, d)$ be the absolute value of the coefficient of the $d$-th deg... | <p>For positive integers $n$ and $m$, we define two polynomials $F_n(x) = x^n$ and $G_m(x) = (x-1)^m$.<br/>
We also define a polynomial $R_{n,m}(x)$ as the remainder of the division of $F_n(x)$ by $G_m(x)$.<br/>
For example, $R_{6,3}(x) = 15x^2 - 24x + 10$.</p>
<p>Let $C(n, m, d)$ be the absolute value of the coefficie... | 472294837 | Sunday, 18th January 2015, 07:00 am | 578 | 40% | medium |
915 | Giant GCDs | The function $s(n)$ is defined recursively for positive integers by
$s(1) = 1$ and $s(n+1) = \big(s(n) - 1\big)^3 +2$ for $n\geq 1$.
The sequence begins: $s(1) = 1, s(2) = 2, s(3) = 3, s(4) = 10, \ldots$.
For positive integers $N$, define $$T(N) = \sum_{a=1}^N \sum_{b=1}^N \gcd\Big(s\big(s(a)\big), s\big(s(b)\big)\B... | The function $s(n)$ is defined recursively for positive integers by
$s(1) = 1$ and $s(n+1) = \big(s(n) - 1\big)^3 +2$ for $n\geq 1$.
The sequence begins: $s(1) = 1, s(2) = 2, s(3) = 3, s(4) = 10, \ldots$.
For positive integers $N$, define $$T(N) = \sum_{a=1}^N \sum_{b=1}^N \gcd\Big(s\big(s(a)\big), s\big(s(b)\big)\B... | <p>
The function $s(n)$ is defined recursively for positive integers by
$s(1) = 1$ and $s(n+1) = \big(s(n) - 1\big)^3 +2$ for $n\geq 1$.<br/>
The sequence begins: $s(1) = 1, s(2) = 2, s(3) = 3, s(4) = 10, \ldots$.</p>
<p>
For positive integers $N$, define $$T(N) = \sum_{a=1}^N \sum_{b=1}^N \gcd\Big(s\big(s(a)\big), s... | 55601924 | Saturday, 2nd November 2024, 07:00 pm | 193 | 45% | medium |
520 | Simbers | We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.
For example, $141221242$ is a $9$-digit simber because it has three $1$'s, four $2$'s and two $4$'s.
Let $Q(n)$ be the count of all simbers with... | We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.
For example, $141221242$ is a $9$-digit simber because it has three $1$'s, four $2$'s and two $4$'s.
Let $Q(n)$ be the count of all simbers with... | <p>We define a <dfn>simber</dfn> to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.</p>
<p>For example, $141221242$ is a $9$-digit simber because it has three $1$'s, four $2$'s and two $4$'s. </p>
<p>Let $Q(n)$ be t... | 238413705 | Saturday, 13th June 2015, 10:00 pm | 460 | 45% | medium |
691 | Long Substring with Many Repetitions | Given a character string $s$, we define $L(k,s)$ to be the length of the longest substring of $s$ which appears at least $k$ times in $s$, or $0$ if such a substring does not exist. For example, $L(3,\text{“bbabcabcabcacba”})=4$ because of the three occurrences of the substring $\text{“abca”}$, and $L(2,\text{“bbabcabc... | Given a character string $s$, we define $L(k,s)$ to be the length of the longest substring of $s$ which appears at least $k$ times in $s$, or $0$ if such a substring does not exist. For example, $L(3,\text{“bbabcabcabcacba”})=4$ because of the three occurrences of the substring $\text{“abca”}$, and $L(2,\text{“bbabcabc... | <p>Given a character string $s$, we define $L(k,s)$ to be the length of the longest substring of $s$ which appears at least $k$ times in $s$, or $0$ if such a substring does not exist. For example, $L(3,\text{“bbabcabcabcacba”})=4$ because of the three occurrences of the substring $\text{“abca”}$, and $L(2,\text{“bbabc... | 11570761 | Sunday, 1st December 2019, 10:00 am | 297 | 40% | medium |
626 | Counting Binary Matrices | A binary matrix is a matrix consisting entirely of $0$s and $1$s. Consider the following transformations that can be performed on a binary matrix:
Swap any two rows
Swap any two columns
Flip all elements in a single row ($1$s become $0$s, $0$s become $1$s)
Flip all elements in a single column
Two binary matrices $A$ ... | A binary matrix is a matrix consisting entirely of $0$s and $1$s. Consider the following transformations that can be performed on a binary matrix:
Swap any two rows
Swap any two columns
Flip all elements in a single row ($1$s become $0$s, $0$s become $1$s)
Flip all elements in a single column
Two binary matrices $A$ ... | <p>A binary matrix is a matrix consisting entirely of $0$s and $1$s. Consider the following transformations that can be performed on a binary matrix:</p>
<ul>
<li>Swap any two rows</li>
<li>Swap any two columns</li>
<li>Flip all elements in a single row ($1$s become $0$s, $0$s become $1$s)</li>
<li>Flip all elements in... | 695577663 | Saturday, 5th May 2018, 07:00 pm | 253 | 70% | hard |
885 | Sorted Digits | For a positive integer $d$, let $f(d)$ be the number created by sorting the digits of $d$ in ascending order, removing any zeros. For example, $f(3403) = 334$.
Let $S(n)$ be the sum of $f(d)$ for all positive integers $d$ of $n$ digits or less. You are given $S(1) = 45$ and $S(5) = 1543545675$.
Find $S(18)$. Give you... | For a positive integer $d$, let $f(d)$ be the number created by sorting the digits of $d$ in ascending order, removing any zeros. For example, $f(3403) = 334$.
Let $S(n)$ be the sum of $f(d)$ for all positive integers $d$ of $n$ digits or less. You are given $S(1) = 45$ and $S(5) = 1543545675$.
Find $S(18)$. Give you... | <p>
For a positive integer $d$, let $f(d)$ be the number created by sorting the digits of $d$ in ascending order, removing any zeros. For example, $f(3403) = 334$.</p>
<p>
Let $S(n)$ be the sum of $f(d)$ for all positive integers $d$ of $n$ digits or less. You are given $S(1) = 45$ and $S(5) = 1543545675$.</p>
<p>
Find... | 827850196 | Sunday, 7th April 2024, 11:00 am | 899 | 10% | easy |
538 | Maximum Quadrilaterals | Consider a positive integer sequence $S = (s_1, s_2, \dots, s_n)$.
Let $f(S)$ be the perimeter of the maximum-area quadrilateral whose side lengths are $4$ elements $(s_i, s_j, s_k, s_l)$ of $S$ (all $i, j, k, l$ distinct). If there are many quadrilaterals with the same maximum area, then choose the one with the larges... | Consider a positive integer sequence $S = (s_1, s_2, \dots, s_n)$.
Let $f(S)$ be the perimeter of the maximum-area quadrilateral whose side lengths are $4$ elements $(s_i, s_j, s_k, s_l)$ of $S$ (all $i, j, k, l$ distinct). If there are many quadrilaterals with the same maximum area, then choose the one with the larges... | <p>Consider a positive integer sequence $S = (s_1, s_2, \dots, s_n)$.</p>
<p>Let $f(S)$ be the perimeter of the maximum-area quadrilateral whose side lengths are $4$ elements $(s_i, s_j, s_k, s_l)$ of $S$ (all $i, j, k, l$ distinct). If there are many quadrilaterals with the same maximum area, then choose the one with ... | 22472871503401097 | Sunday, 13th December 2015, 01:00 am | 360 | 40% | medium |
605 | Pairwise Coin-Tossing Game | Consider an $n$-player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and so on and so forth, all the way up to round $n$, which takes place between players $n$ and $1$. Then round $n+1$ takes place between players $1$ and $2$ as t... | Consider an $n$-player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and so on and so forth, all the way up to round $n$, which takes place between players $n$ and $1$. Then round $n+1$ takes place between players $1$ and $2$ as t... | <p>Consider an $n$-player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and so on and so forth, all the way up to round $n$, which takes place between players $n$ and $1$. Then round $n+1$ takes place between players $1$ and $2$ a... | 59992576 | Sunday, 28th May 2017, 07:00 am | 938 | 25% | easy |
359 | Hilbert's New Hotel | An infinite number of people (numbered $1$, $2$, $3$, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered $1$, $2$, $3$, etc.), and each floor contains an infinite number of rooms (numbered $1$, $2$, $3$, etc.).
Initially the hotel is empty. ... | An infinite number of people (numbered $1$, $2$, $3$, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered $1$, $2$, $3$, etc.), and each floor contains an infinite number of rooms (numbered $1$, $2$, $3$, etc.).
Initially the hotel is empty. ... | <p>
An infinite number of people (numbered $1$, $2$, $3$, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered $1$, $2$, $3$, etc.), and each floor contains an infinite number of rooms (numbered $1$, $2$, $3$, etc.).
</p>
<p>
Initially the hotel... | 40632119 | Saturday, 19th November 2011, 10:00 pm | 1694 | 25% | easy |
657 | Incomplete Words | In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete w... | In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete w... | <p>In the context of <strong>formal languages</strong>, any finite sequence of letters of a given <strong>alphabet</strong> $\Sigma$ is called a <strong>word</strong> over $\Sigma$. We call a word <dfn>incomplete</dfn> if it does not contain every letter of $\Sigma$.</p>
<p>
For example, using the alphabet $\Sigma=\{ a... | 219493139 | Saturday, 23rd February 2019, 01:00 pm | 618 | 30% | easy |
490 | Jumping Frog | There are $n$ stones in a pond, numbered $1$ to $n$. Consecutive stones are spaced one unit apart.
A frog sits on stone $1$. He wishes to visit each stone exactly once, stopping on stone $n$. However, he can only jump from one stone to another if they are at most $3$ units apart. In other words, from stone $i$, he can ... | There are $n$ stones in a pond, numbered $1$ to $n$. Consecutive stones are spaced one unit apart.
A frog sits on stone $1$. He wishes to visit each stone exactly once, stopping on stone $n$. However, he can only jump from one stone to another if they are at most $3$ units apart. In other words, from stone $i$, he can ... | <p>There are $n$ stones in a pond, numbered $1$ to $n$. Consecutive stones are spaced one unit apart.</p>
<p>A frog sits on stone $1$. He wishes to visit each stone exactly once, stopping on stone $n$. However, he can only jump from one stone to another if they are at most $3$ units apart. In other words, from stone $i... | 777577686 | Sunday, 23rd November 2014, 07:00 am | 351 | 95% | hard |
234 | Semidivisible Numbers | For an integer $n \ge 4$, we define the lower prime square root of $n$, denoted by $\operatorname{lps}(n)$, as the largest prime $\le \sqrt n$ and the upper prime square root of $n$, $\operatorname{ups}(n)$, as the smallest prime $\ge \sqrt n$.
So, for example, $\operatorname{lps}(4) = 2 = \operatorname{ups}(4)$, $\ope... | For an integer $n \ge 4$, we define the lower prime square root of $n$, denoted by $\operatorname{lps}(n)$, as the largest prime $\le \sqrt n$ and the upper prime square root of $n$, $\operatorname{ups}(n)$, as the smallest prime $\ge \sqrt n$.
So, for example, $\operatorname{lps}(4) = 2 = \operatorname{ups}(4)$, $\ope... | <p>For an integer $n \ge 4$, we define the <dfn>lower prime square root</dfn> of $n$, denoted by $\operatorname{lps}(n)$, as the largest prime $\le \sqrt n$ and the <dfn>upper prime square root</dfn> of $n$, $\operatorname{ups}(n)$, as the smallest prime $\ge \sqrt n$.</p>
<p>So, for example, $\operatorname{lps}(4) = 2... | 1259187438574927161 | Saturday, 28th February 2009, 01:00 am | 3538 | 50% | medium |
110 | Diophantine Reciprocals II | In the following equation $x$, $y$, and $n$ are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$
It can be verified that when $n = 1260$ there are $113$ distinct solutions and this is the least value of $n$ for which the total number of distinct solutions exceeds one hundred.
What is the least value of... | In the following equation $x$, $y$, and $n$ are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$
It can be verified that when $n = 1260$ there are $113$ distinct solutions and this is the least value of $n$ for which the total number of distinct solutions exceeds one hundred.
What is the least value of... | <p>In the following equation $x$, $y$, and $n$ are positive integers.</p>
<p>$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$</p>
<p>It can be verified that when $n = 1260$ there are $113$ distinct solutions and this is the least value of $n$ for which the total number of distinct solutions exceeds one hundred.</p>
<p>Wh... | 9350130049860600 | Friday, 2nd December 2005, 06:00 pm | 9144 | 40% | medium |
857 | Beautiful Graphs | A graph is made up of vertices and coloured edges.
Between every two distinct vertices there must be exactly one of the following:
A red directed edge one way, and a blue directed edge the other way
A green undirected edge
A brown undirected edge
Such a graph is called beautiful if
A cycle of edges contains a re... | A graph is made up of vertices and coloured edges.
Between every two distinct vertices there must be exactly one of the following:
A red directed edge one way, and a blue directed edge the other way
A green undirected edge
A brown undirected edge
Such a graph is called beautiful if
A cycle of edges contains a re... | <p>
A graph is made up of vertices and coloured edges.
Between every two distinct vertices there must be exactly one of the following:</p>
<ul>
<li>A red directed edge one way, and a blue directed edge the other way</li>
<li>A green undirected edge</li>
<li>A brown undirected edge</li>
</ul>
Such a graph is called <i... | 966332096 | Sunday, 1st October 2023, 02:00 am | 188 | 60% | hard |
413 | One-child Numbers | We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$.
For example, $5671$ is a $4$-digit one-child number. Among all its sub-strings $5$, $6$, $7$, $1$, $56$, $67$, $71$, $567$, $671$ and $5671$, only $56$ is divisible by $4$.
Similarly, ... | We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$.
For example, $5671$ is a $4$-digit one-child number. Among all its sub-strings $5$, $6$, $7$, $1$, $56$, $67$, $71$, $567$, $671$ and $5671$, only $56$ is divisible by $4$.
Similarly, ... | <p>We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$.</p>
<p>For example, $5671$ is a $4$-digit one-child number. Among all its sub-strings $5$, $6$, $7$, $1$, $56$, $67$, $71$, $567$, $671$ and $5671$, only $56$ is divisible by $4$.<b... | 3079418648040719 | Sunday, 3rd February 2013, 04:00 am | 440 | 75% | hard |
292 | Pythagorean Polygons | We shall define a pythagorean polygon to be a convex polygon with the following properties:there are at least three vertices,
no three vertices are aligned,
each vertex has integer coordinates,
each edge has integer length.For a given integer $n$, define $P(n)$ as the number of distinct pythagorean polygons for which ... | We shall define a pythagorean polygon to be a convex polygon with the following properties:there are at least three vertices,
no three vertices are aligned,
each vertex has integer coordinates,
each edge has integer length.For a given integer $n$, define $P(n)$ as the number of distinct pythagorean polygons for which ... | <p>We shall define a <dfn>pythagorean polygon</dfn> to be a <strong>convex polygon</strong> with the following properties:<br/></p><ul><li>there are at least three vertices,</li>
<li>no three vertices are aligned,</li>
<li>each vertex has <b>integer coordinates</b>,</li>
<li>each edge has <b>integer length</b>.</li></... | 3600060866 | Saturday, 15th May 2010, 01:00 am | 640 | 65% | hard |
182 | RSA Encryption | The RSA encryption is based on the following procedure:
Generate two distinct primes $p$ and $q$.Compute $n = pq$ and $\phi = (p - 1)(q - 1)$.
Find an integer $e$, $1 \lt e \lt \phi$, such that $\gcd(e, \phi) = 1$.
A message in this system is a number in the interval $[0, n - 1]$.
A text to be encrypted is then somehow... | The RSA encryption is based on the following procedure:
Generate two distinct primes $p$ and $q$.Compute $n = pq$ and $\phi = (p - 1)(q - 1)$.
Find an integer $e$, $1 \lt e \lt \phi$, such that $\gcd(e, \phi) = 1$.
A message in this system is a number in the interval $[0, n - 1]$.
A text to be encrypted is then somehow... | <p>The RSA encryption is based on the following procedure:</p>
<p>Generate two distinct primes $p$ and $q$.<br/>Compute $n = pq$ and $\phi = (p - 1)(q - 1)$.<br/>
Find an integer $e$, $1 \lt e \lt \phi$, such that $\gcd(e, \phi) = 1$.</p>
<p>A message in this system is a number in the interval $[0, n - 1]$.<br/>
A text... | 399788195976 | Friday, 15th February 2008, 01:00 pm | 2923 | 60% | hard |
588 | Quintinomial Coefficients | The coefficients in the expansion of $(x+1)^k$ are called binomial coefficients.
Analoguously the coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$ are called quintinomial coefficients. (quintus= Latin for fifth).
Consider the expansion of $(x^4+x^3+x^2+x+1)^3$:
$x^{12}+3x^{11}+6x^{10}+10x^9+15x^8+18x^7+19x^6+18... | The coefficients in the expansion of $(x+1)^k$ are called binomial coefficients.
Analoguously the coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$ are called quintinomial coefficients. (quintus= Latin for fifth).
Consider the expansion of $(x^4+x^3+x^2+x+1)^3$:
$x^{12}+3x^{11}+6x^{10}+10x^9+15x^8+18x^7+19x^6+18... | <p>
The coefficients in the expansion of $(x+1)^k$ are called <strong>binomial coefficients</strong>.<br/>
Analoguously the coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$ are called <strong>quintinomial coefficients</strong>.<br/> (quintus= Latin for fifth).
</p>
<p>
Consider the expansion of $(x^4+x^3+x^2+x+1)... | 11651930052 | Sunday, 29th January 2017, 04:00 am | 494 | 40% | medium |
766 | Sliding Block Puzzle | A sliding block puzzle is a puzzle where pieces are confined to a grid and by sliding the pieces a final configuration is reached. In this problem the pieces can only be slid in multiples of one unit in the directions up, down, left, right.
A reachable configuration is any arrangement of the pieces that can be achieved... | A sliding block puzzle is a puzzle where pieces are confined to a grid and by sliding the pieces a final configuration is reached. In this problem the pieces can only be slid in multiples of one unit in the directions up, down, left, right.
A reachable configuration is any arrangement of the pieces that can be achieved... | <p>A <strong>sliding block puzzle</strong> is a puzzle where pieces are confined to a grid and by sliding the pieces a final configuration is reached. In this problem the pieces can only be slid in multiples of one unit in the directions up, down, left, right.</p>
<p>A <dfn>reachable configuration</dfn> is any arrangem... | 2613742 | Saturday, 2nd October 2021, 11:00 pm | 350 | 35% | medium |
406 | Guessing Game | We are trying to find a hidden number selected from the set of integers $\{1, 2, \dots, n\}$ by asking questions.
Each number (question) we ask, we get one of three possible answers: "Your guess is lower than the hidden number" (and you incur a cost of $a$), or
"Your guess is higher than the hidden number" (and you i... | We are trying to find a hidden number selected from the set of integers $\{1, 2, \dots, n\}$ by asking questions.
Each number (question) we ask, we get one of three possible answers: "Your guess is lower than the hidden number" (and you incur a cost of $a$), or
"Your guess is higher than the hidden number" (and you i... | <p>We are trying to find a hidden number selected from the set of integers $\{1, 2, \dots, n\}$ by asking questions.
Each number (question) we ask, we get one of three possible answers:<br/></p><ul><li> "Your guess is lower than the hidden number" (and you incur a cost of $a$), or</li>
<li> "Your guess is higher than ... | 36813.12757207 | Sunday, 16th December 2012, 07:00 am | 430 | 50% | medium |
392 | Enmeshed Unit Circle | A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.
Consider rectilinear grids in the Cartesian coordinate system with the following properties:The gridlines are parallel to the axes of the Cartesian coor... | A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.
Consider rectilinear grids in the Cartesian coordinate system with the following properties:The gridlines are parallel to the axes of the Cartesian coor... | <p>
A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.<br/>
An example of such grid is logarithmic graph paper.
</p>
<p>
Consider rectilinear grids in the Cartesian coordinate system with the following properties:<br/></p><ul><li>The gridlines are parallel ... | 3.1486734435 | Saturday, 1st September 2012, 02:00 pm | 878 | 35% | medium |
166 | Criss Cross | A $4 \times 4$ grid is filled with digits $d$, $0 \le d \le 9$.
It can be seen that in the grid
\begin{matrix}
6 & 3 & 3 & 0\\
5 & 0 & 4 & 3\\
0 & 7 & 1 & 4\\
1 & 2 & 4 & 5
\end{matrix}
the sum of each row and each column has the value $12$. Moreover the sum of each diagonal is also $12$.
In how many ways can you fill ... | A $4 \times 4$ grid is filled with digits $d$, $0 \le d \le 9$.
It can be seen that in the grid
\begin{matrix}
6 & 3 & 3 & 0\\
5 & 0 & 4 & 3\\
0 & 7 & 1 & 4\\
1 & 2 & 4 & 5
\end{matrix}
the sum of each row and each column has the value $12$. Moreover the sum of each diagonal is also $12$.
In how many ways can you fill ... | <p>A $4 \times 4$ grid is filled with digits $d$, $0 \le d \le 9$.</p>
<p>It can be seen that in the grid
\begin{matrix}
6 & 3 & 3 & 0\\
5 & 0 & 4 & 3\\
0 & 7 & 1 & 4\\
1 & 2 & 4 & 5
\end{matrix}
the sum of each row and each column has the value $12$. Moreover the sum of ... | 7130034 | Saturday, 3rd November 2007, 01:00 pm | 4570 | 50% | medium |
450 | Hypocycloid and Lattice Points | A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
$$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$$
$$y(t) = (R - r) \sin(t) - r \sin(\frac {R - r} r t)$$
... | A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
$$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$$
$$y(t) = (R - r) \sin(t) - r \sin(\frac {R - r} r t)$$
... | <p>A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:</p>
<p>$$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$$
$$y(t) = (R - r) \sin(t) - r \sin(\frac {R - ... | 583333163984220940 | Sunday, 15th December 2013, 07:00 am | 217 | 100% | hard |
774 | Conjunctive Sequences | Let '$\&$' denote the bitwise AND operation.
For example, $10\,\&\, 12 = 1010_2\,\&\, 1100_2 = 1000_2 = 8$.
We shall call a finite sequence of non-negative integers $(a_1, a_2, \ldots, a_n)$ conjunctive if $a_i\,\&\, a_{i+1} \neq 0$ for all $i=1\ldots n-1$.
Define $c(n,b)$ to be the number of conjunctive sequences of l... | Let '$\&$' denote the bitwise AND operation.
For example, $10\,\&\, 12 = 1010_2\,\&\, 1100_2 = 1000_2 = 8$.
We shall call a finite sequence of non-negative integers $(a_1, a_2, \ldots, a_n)$ conjunctive if $a_i\,\&\, a_{i+1} \neq 0$ for all $i=1\ldots n-1$.
Define $c(n,b)$ to be the number of conjunctive sequences of l... | <p>Let '$\&$' denote the bitwise AND operation.<br/>
For example, $10\,\&\, 12 = 1010_2\,\&\, 1100_2 = 1000_2 = 8$.</p>
<p>We shall call a finite sequence of non-negative integers $(a_1, a_2, \ldots, a_n)$ <dfn>conjunctive</dfn> if $a_i\,\&\, a_{i+1} \neq 0$ for all $i=1\ldots n-1$.</p>
<p>Define $c(n,b... | 459155763 | Saturday, 27th November 2021, 10:00 pm | 154 | 90% | hard |
210 | Obtuse Angled Triangles | Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$.
Let $O$ be the point $(0,0)$ and $C$ the point $(r/4,r/4)$.
Let $N(r)$ be the number of points $B$ in $S(r)$, so that the triangle $OBC$ has an obtuse angle, i.e. the largest angle $\alpha$ satisfies $90^\circ \lt \alpha \... | Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$.
Let $O$ be the point $(0,0)$ and $C$ the point $(r/4,r/4)$.
Let $N(r)$ be the number of points $B$ in $S(r)$, so that the triangle $OBC$ has an obtuse angle, i.e. the largest angle $\alpha$ satisfies $90^\circ \lt \alpha \... | Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$.<br/>
Let $O$ be the point $(0,0)$ and $C$ the point $(r/4,r/4)$. <br/>
Let $N(r)$ be the number of points $B$ in $S(r)$, so that the triangle $OBC$ has an obtuse angle, i.e. the largest angle $\alpha$ satisfies $90^\circ \l... | 1598174770174689458 | Friday, 26th September 2008, 10:00 pm | 1909 | 70% | hard |
402 | Integer-valued Polynomials | It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of $6$ for every integer $n$. It can also be shown that $6$ is the largest integer satisfying this property.
Define $M(a, b, c)$ as the maximum $m$ such that $n^4 + an^3 + bn^2 + cn$ is a multiple of $m$ for all integers $n$. For example, $M(4... | It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of $6$ for every integer $n$. It can also be shown that $6$ is the largest integer satisfying this property.
Define $M(a, b, c)$ as the maximum $m$ such that $n^4 + an^3 + bn^2 + cn$ is a multiple of $m$ for all integers $n$. For example, $M(4... | <p>
It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of $6$ for every integer $n$. It can also be shown that $6$ is the largest integer satisfying this property.
</p>
<p>
Define $M(a, b, c)$ as the maximum $m$ such that $n^4 + an^3 + bn^2 + cn$ is a multiple of $m$ for all integers $n$. For ex... | 356019862 | Saturday, 17th November 2012, 07:00 pm | 467 | 55% | medium |
650 | Divisors of Binomial Product | Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients.
For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose 2} \times {5 \choose 3} \times {5 \choose 4} \times {5 \choose 5} = 1 \times 5 \times 10 \times 10 \times 5 \times 1 = 2500$.
Let $D(n) = \displaystyle... | Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients.
For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose 2} \times {5 \choose 3} \times {5 \choose 4} \times {5 \choose 5} = 1 \times 5 \times 10 \times 10 \times 5 \times 1 = 2500$.
Let $D(n) = \displaystyle... | <p>
Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients.<br>
For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose 2} \times {5 \choose 3} \times {5 \choose 4} \times {5 \choose 5} = 1 \times 5 \times 10 \times 10 \times 5 \times 1 = 2500$.
</br></p>
<p>
Let $... | 538319652 | Saturday, 5th January 2019, 04:00 pm | 1862 | 10% | easy |
474 | Last Digits of Divisors | For a positive integer $n$ and digits $d$, we define $F(n, d)$ as the number of the divisors of $n$ whose last digits equal $d$.
For example, $F(84, 4) = 3$. Among the divisors of $84$ ($1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$), three of them ($4, 14, 84$) have the last digit $4$.
We can also verify that $F(12!, 12... | For a positive integer $n$ and digits $d$, we define $F(n, d)$ as the number of the divisors of $n$ whose last digits equal $d$.
For example, $F(84, 4) = 3$. Among the divisors of $84$ ($1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$), three of them ($4, 14, 84$) have the last digit $4$.
We can also verify that $F(12!, 12... | <p>
For a positive integer $n$ and digits $d$, we define $F(n, d)$ as the number of the divisors of $n$ whose last digits equal $d$.<br/>
For example, $F(84, 4) = 3$. Among the divisors of $84$ ($1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$), three of them ($4, 14, 84$) have the last digit $4$.
</p>
<p>
We can also verify... | 9690646731515010 | Sunday, 1st June 2014, 07:00 am | 467 | 50% | medium |
150 | Sub-triangle Sums | In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest possible.
In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of −42.
We wish to make such a triangular array ... | In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest possible.
In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of −42.
We wish to make such a triangular array ... | <p>In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest possible.</p>
<p>In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of −42.</p>
<div class="center">
<img al... | -271248680 | Friday, 13th April 2007, 10:00 pm | 4436 | 55% | medium |
887 | Bounded Binary Search | Consider the problem of determining a secret number from a set $\{1, ..., N\}$ by repeatedly choosing a number $y$ and asking "Is the secret number greater than $y$?".
If $N=1$ then no questions need to be asked. If $N=2$ then only one question needs to be asked. If $N=64$ then six questions need to be asked. However, ... | Consider the problem of determining a secret number from a set $\{1, ..., N\}$ by repeatedly choosing a number $y$ and asking "Is the secret number greater than $y$?".
If $N=1$ then no questions need to be asked. If $N=2$ then only one question needs to be asked. If $N=64$ then six questions need to be asked. However, ... | <p>Consider the problem of determining a secret number from a set $\{1, ..., N\}$ by repeatedly choosing a number $y$ and asking "Is the secret number greater than $y$?".</p>
<p>If $N=1$ then no questions need to be asked. If $N=2$ then only one question needs to be asked. If $N=64$ then six questions need to be asked.... | 39896187138661622 | Saturday, 20th April 2024, 05:00 pm | 253 | 30% | easy |
71 | Ordered Fractions | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \f... | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \f... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \fr... | 428570 | Friday, 4th June 2004, 06:00 pm | 32418 | 10% | easy |
256 | Tatami-Free Rooms | Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
Assuming that the only type of available tatami has dimensions $1 \times 2$, there are obviously some limitations for the shape and size of the rooms that can be covered.
For this problem, we consider only rectangular rooms with... | Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
Assuming that the only type of available tatami has dimensions $1 \times 2$, there are obviously some limitations for the shape and size of the rooms that can be covered.
For this problem, we consider only rectangular rooms with... | <p>Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.</p>
<p>Assuming that the only type of available tatami has dimensions $1 \times 2$, there are obviously some limitations for the shape and size of the rooms that can be covered.</p>
<p>For this problem, we consider only recta... | 85765680 | Saturday, 19th September 2009, 01:00 am | 830 | 80% | hard |
38 | Pandigital Multiples | Take the number $192$ and multiply it by each of $1$, $2$, and $3$:
\begin{align}
192 \times 1 &= 192\\
192 \times 2 &= 384\\
192 \times 3 &= 576
\end{align}
By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ and $(1,2,3)$.
The same ca... | Take the number $192$ and multiply it by each of $1$, $2$, and $3$:
\begin{align}
192 \times 1 &= 192\\
192 \times 2 &= 384\\
192 \times 3 &= 576
\end{align}
By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ and $(1,2,3)$.
The same ca... | <p>Take the number $192$ and multiply it by each of $1$, $2$, and $3$:</p>
\begin{align}
192 \times 1 &= 192\\
192 \times 2 &= 384\\
192 \times 3 &= 576
\end{align}
<p>By concatenating each product we get the $1$ to $9$ pandigital, $192384576$. We will call $192384576$ the concatenated product of $192$ and $(1,2,3)$.</... | 932718654 | Friday, 28th February 2003, 06:00 pm | 69062 | 5% | easy |
429 | Sum of Squares of Unitary Divisors | A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$.
The unitary divisors of $4! = 24$ are $1, 3, 8$ and $24$.
The sum of their squares is $1^2 + 3^2 + 8^2 + 24^2 = 650$.
Let $S(n)$ represent the sum of the squares of the unitary divisors of $n$. Thus $S(4!)=650$.
Find... | A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$.
The unitary divisors of $4! = 24$ are $1, 3, 8$ and $24$.
The sum of their squares is $1^2 + 3^2 + 8^2 + 24^2 = 650$.
Let $S(n)$ represent the sum of the squares of the unitary divisors of $n$. Thus $S(4!)=650$.
Find... | <p>
A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$.<br/>
The unitary divisors of $4! = 24$ are $1, 3, 8$ and $24$.<br/>
The sum of their squares is $1^2 + 3^2 + 8^2 + 24^2 = 650$.
</p>
<p>
Let $S(n)$ represent the sum of the squares of the unitary divisors of $n$. Thu... | 98792821 | Sunday, 26th May 2013, 04:00 am | 2796 | 20% | easy |
567 | Reciprocal Games I | Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights is turned on with the probability of $\frac 1 2$, independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use h... | Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights is turned on with the probability of $\frac 1 2$, independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use h... | <p>Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights is turned on with the probability of $\frac 1 2$, independently of its former state or the state of the other light bulbs.</p>
<p>While discussing with his friend Jerry ho... | 75.44817535 | Saturday, 3rd September 2016, 04:00 pm | 331 | 55% | medium |
319 | Bounded Sequences | Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that:
$x_1 = 2$
for all $1 \lt i \le n$: $x_{i - 1} \lt x_i$
for all $i$ and $j$ with $1 \le i, j \le n$: $(x_i)^j \lt (x_j + 1)^i$.
There are only five such sequences of length $2$, namely:
$\{2,4\}$, $\{2,5\}$, $\{2,6\}$, $\{2,7\}$ and $\{2,8\}$.
There are ... | Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that:
$x_1 = 2$
for all $1 \lt i \le n$: $x_{i - 1} \lt x_i$
for all $i$ and $j$ with $1 \le i, j \le n$: $(x_i)^j \lt (x_j + 1)^i$.
There are only five such sequences of length $2$, namely:
$\{2,4\}$, $\{2,5\}$, $\{2,6\}$, $\{2,7\}$ and $\{2,8\}$.
There are ... | <p>
Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that:
</p><ul><li>$x_1 = 2$</li>
<li>for all $1 \lt i \le n$: $x_{i - 1} \lt x_i$</li>
<li>for all $i$ and $j$ with $1 \le i, j \le n$: $(x_i)^j \lt (x_j + 1)^i$.</li>
</ul><p>
There are only five such sequences of length $2$, namely:
$\{2,4\}$, $\{2,5\}$,... | 268457129 | Saturday, 8th January 2011, 07:00 pm | 453 | 90% | hard |
701 | Random Connected Area | Consider a rectangle made up of $W \times H$ square cells each with area $1$. Each cell is independently coloured black with probability $0.5$ otherwise white. Black cells sharing an edge are assumed to be connected.Consider the maximum area of connected cells.
Define $E(W,H)$ to be the expected value of this maximum ... | Consider a rectangle made up of $W \times H$ square cells each with area $1$. Each cell is independently coloured black with probability $0.5$ otherwise white. Black cells sharing an edge are assumed to be connected.Consider the maximum area of connected cells.
Define $E(W,H)$ to be the expected value of this maximum ... | <p>
Consider a rectangle made up of $W \times H$ square cells each with area $1$.<br/> Each cell is independently coloured black with probability $0.5$ otherwise white. Black cells sharing an edge are assumed to be connected.<br/>Consider the maximum area of connected cells.</p>
<p>
Define $E(W,H)$ to be the expected v... | 13.51099836 | Saturday, 8th February 2020, 04:00 pm | 388 | 40% | medium |
323 | Bitwise-OR Operations on Random Integers | Let $y_0, y_1, y_2, \dots$ be a sequence of random unsigned $32$-bit integers
(i.e. $0 \le y_i \lt 2^{32}$, every value equally likely).
For the sequence $x_i$ the following recursion is given:$x_0 = 0$ and
$x_i = x_{i - 1} \boldsymbol \mid y_{i - 1}$, for $i \gt 0$. ($\boldsymbol \mid$ is the bitwise-OR operator).
It ... | Let $y_0, y_1, y_2, \dots$ be a sequence of random unsigned $32$-bit integers
(i.e. $0 \le y_i \lt 2^{32}$, every value equally likely).
For the sequence $x_i$ the following recursion is given:$x_0 = 0$ and
$x_i = x_{i - 1} \boldsymbol \mid y_{i - 1}$, for $i \gt 0$. ($\boldsymbol \mid$ is the bitwise-OR operator).
It ... | <p>Let $y_0, y_1, y_2, \dots$ be a sequence of random unsigned $32$-bit integers<br/>
(i.e. $0 \le y_i \lt 2^{32}$, every value equally likely).</p>
<p>For the sequence $x_i$ the following recursion is given:<br/></p><ul><li>$x_0 = 0$ and</li>
<li>$x_i = x_{i - 1} \boldsymbol \mid y_{i - 1}$, for $i \gt 0$. ($\boldsymb... | 6.3551758451 | Sunday, 6th February 2011, 07:00 am | 4272 | 20% | easy |
869 | Prime Guessing | A prime is drawn uniformly from all primes not exceeding $N$. The prime is written in binary notation, and a player tries to guess it bit-by-bit starting at the least significant bit. The player scores one point for each bit they guess correctly. Immediately after each guess, the player is informed whether their guess ... | A prime is drawn uniformly from all primes not exceeding $N$. The prime is written in binary notation, and a player tries to guess it bit-by-bit starting at the least significant bit. The player scores one point for each bit they guess correctly. Immediately after each guess, the player is informed whether their guess ... | <p>
A prime is drawn uniformly from all primes not exceeding $N$. The prime is written in binary notation, and a player tries to guess it bit-by-bit starting at the least significant bit. The player scores one point for each bit they guess correctly. Immediately after each guess, the player is informed whether their gu... | 14.97696693 | Saturday, 23rd December 2023, 01:00 pm | 758 | 15% | easy |
877 | XOR-Equation A | 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_... | 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_... | <p>
We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.<br/>
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 b... | 336785000760344621 | Saturday, 17th February 2024, 01:00 pm | 433 | 20% | easy |
159 | Digital Root Sums of Factorisations | A composite number can be factored many different ways.
For instance, not including multiplication by one, $24$ can be factored in $7$ distinct ways:
\begin{align}
24 &= 2 \times 2 \times 2 \times 3\\
24 &= 2 \times 3 \times 4\\
24 &= 2 \times 2 \times 6\\
24 &= 4 \times 6\\
24 &= 3 \times 8\\
24 &= 2 \times 12\\
24 ... | A composite number can be factored many different ways.
For instance, not including multiplication by one, $24$ can be factored in $7$ distinct ways:
\begin{align}
24 &= 2 \times 2 \times 2 \times 3\\
24 &= 2 \times 3 \times 4\\
24 &= 2 \times 2 \times 6\\
24 &= 4 \times 6\\
24 &= 3 \times 8\\
24 &= 2 \times 12\\
24 ... | <p>A composite number can be factored many different ways.
For instance, not including multiplication by one, $24$ can be factored in $7$ distinct ways:</p>
\begin{align}
24 &= 2 \times 2 \times 2 \times 3\\
24 &= 2 \times 3 \times 4\\
24 &= 2 \times 2 \times 6\\
24 &= 4 \times 6\\
24 &= 3 \times 8\\
24 &= 2 \times 1... | 14489159 | Saturday, 30th June 2007, 02:00 pm | 3670 | 60% | hard |
928 | Cribbage | This problem is based on (but not identical to) the scoring for the card game
Cribbage.
Consider a normal pack of $52$ cards. A Hand is a selection of one or more of these cards.
For each Hand the Hand score is the sum of the values of the cards in the Hand where the value of Aces is $1$ and the value of court cards... | This problem is based on (but not identical to) the scoring for the card game
Cribbage.
Consider a normal pack of $52$ cards. A Hand is a selection of one or more of these cards.
For each Hand the Hand score is the sum of the values of the cards in the Hand where the value of Aces is $1$ and the value of court cards... | <p>This problem is based on (but not identical to) the scoring for the card game
<a href="https://en.wikipedia.org/wiki/Cribbage">Cribbage</a>.</p>
<p>
Consider a normal pack of $52$ cards. A Hand is a selection of one or more of these cards.</p>
<p>
For each Hand the <i>Hand score</i> is the sum of the values of the ... | 81108001093 | Sunday, 19th January 2025, 04:00 am | 198 | 35% | medium |
96 | Su Doku | Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) ... | Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) ... | <p>Su Doku (Japanese meaning <i>number place</i>) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (... | 24702 | Friday, 27th May 2005, 06:00 pm | 19168 | 25% | easy |
349 | Langton's Ant | An ant moves on a regular grid of squares that are coloured either black or white.
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:
- if it is on a black square, it flips the colour of the square to white, ... | An ant moves on a regular grid of squares that are coloured either black or white.
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:
- if it is on a black square, it flips the colour of the square to white, ... | <p>
An ant moves on a regular grid of squares that are coloured either black or white.<br/>
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:<br/>
- if it is on a black square, it flips the colour of the squ... | 115384615384614952 | Saturday, 3rd September 2011, 04:00 pm | 2123 | 35% | medium |
488 | Unbalanced Nim | Alice and Bob have enjoyed playing Nim every day. However, they finally got bored of playing ordinary three-heap Nim.
So, they added an extra rule:
- Must not make two heaps of the same size.
The triple $(a, b, c)$ indicates the size of three heaps.
Under this extra rule, $(2,4,5)$ is one of the losing positions for th... | Alice and Bob have enjoyed playing Nim every day. However, they finally got bored of playing ordinary three-heap Nim.
So, they added an extra rule:
- Must not make two heaps of the same size.
The triple $(a, b, c)$ indicates the size of three heaps.
Under this extra rule, $(2,4,5)$ is one of the losing positions for th... | <p>Alice and Bob have enjoyed playing <strong>Nim</strong> every day. However, they finally got bored of playing ordinary three-heap Nim.<br/>
So, they added an extra rule:</p>
<p>- Must not make two heaps of the same size.</p>
<p>The triple $(a, b, c)$ indicates the size of three heaps.<br/>
Under this extra rule, $(2... | 216737278 | Sunday, 9th November 2014, 01:00 am | 264 | 80% | hard |
176 | Common Cathetus Right-angled Triangles | The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to $12$.
Find the smallest integer that can be the length o... | The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to $12$.
Find the smallest integer that can be the length o... | <p>The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to $12$.</p>
<p>Find the smallest integer that can be th... | 96818198400000 | Friday, 4th January 2008, 05:00 pm | 2139 | 70% | hard |
342 | The Totient of a Square Is a Cube | Consider the number $50$.
$50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$. 1
So $2500$ is a square and $\phi(2500)$ is a cube.
Find the sum of all numbers $n$, $1 \lt n \lt 10^{10}$ such that $\phi(n^2)$ is a cube.
1 $\phi$ denotes Euler's totient function. | Consider the number $50$.
$50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$. 1
So $2500$ is a square and $\phi(2500)$ is a cube.
Find the sum of all numbers $n$, $1 \lt n \lt 10^{10}$ such that $\phi(n^2)$ is a cube.
1 $\phi$ denotes Euler's totient function. | <p>
Consider the number $50$.<br/>
$50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$. <sup>1</sup><br/>
So $2500$ is a square and $\phi(2500)$ is a cube.
</p>
<p>
Find the sum of all numbers $n$, $1 \lt n \lt 10^{10}$ such that $\phi(n^2)$ is a cube.
</p>
<p>
<sup>1... | 5943040885644 | Saturday, 11th June 2011, 01:00 pm | 897 | 50% | medium |
734 | A Bit of Prime | The logical-OR of two bits is $0$ if both bits are $0$, otherwise it is $1$.
The bitwise-OR of two positive integers performs a logical-OR operation on each pair of corresponding bits in the binary expansion of its inputs.
For example, the bitwise-OR of $10$ and $6$ is $14$ because $10 = 1010_2$, $6 = 0110_2$ and $14... | The logical-OR of two bits is $0$ if both bits are $0$, otherwise it is $1$.
The bitwise-OR of two positive integers performs a logical-OR operation on each pair of corresponding bits in the binary expansion of its inputs.
For example, the bitwise-OR of $10$ and $6$ is $14$ because $10 = 1010_2$, $6 = 0110_2$ and $14... | <p>
The <strong>logical-OR</strong> of two bits is $0$ if both bits are $0$, otherwise it is $1$.<br/>
The <strong>bitwise-OR</strong> of two positive integers performs a logical-OR operation on each pair of corresponding bits in the binary expansion of its inputs.
</p>
<p>
For example, the bitwise-OR of $10$ and $6$ i... | 557988060 | Saturday, 14th November 2020, 07:00 pm | 363 | 35% | medium |
187 | Semiprimes | A composite is a number containing at least two prime factors. For example, $15 = 3 \times 5$; $9 = 3 \times 3$; $12 = 2 \times 2 \times 3$.
There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors:
$4, 6, 9, 10, 14, 15, 21, 22, 25, 26$.
How many composite integers, $n \lt... | A composite is a number containing at least two prime factors. For example, $15 = 3 \times 5$; $9 = 3 \times 3$; $12 = 2 \times 2 \times 3$.
There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors:
$4, 6, 9, 10, 14, 15, 21, 22, 25, 26$.
How many composite integers, $n \lt... | <p>A composite is a number containing at least two prime factors. For example, $15 = 3 \times 5$; $9 = 3 \times 3$; $12 = 2 \times 2 \times 3$.</p>
<p>There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors:
$4, 6, 9, 10, 14, 15, 21, 22, 25, 26$.</p>
<p>How many composite... | 17427258 | Saturday, 22nd March 2008, 09:00 am | 12184 | 25% | easy |
845 | Prime Digit Sum | Let $D(n)$ be the $n$-th positive integer that has the sum of its digits a prime.
For example, $D(61) = 157$ and $D(10^8) = 403539364$.
Find $D(10^{16})$. | Let $D(n)$ be the $n$-th positive integer that has the sum of its digits a prime.
For example, $D(61) = 157$ and $D(10^8) = 403539364$.
Find $D(10^{16})$. | <p>
Let $D(n)$ be the $n$-th positive integer that has the sum of its digits a prime.<br/>
For example, $D(61) = 157$ and $D(10^8) = 403539364$.</p>
<p>
Find $D(10^{16})$.</p> | 45009328011709400 | Saturday, 27th May 2023, 05:00 pm | 820 | 15% | easy |
264 | Triangle Centres | Consider all the triangles having:
All their vertices on lattice pointsInteger coordinates.
CircumcentreCentre of the circumscribed circle at the origin $O$.
OrthocentrePoint where the three altitudes meet at the point $H(5, 0)$.
There are nine such triangles having a perimeter $\le 50$.
Listed and shown in ascending o... | Consider all the triangles having:
All their vertices on lattice pointsInteger coordinates.
CircumcentreCentre of the circumscribed circle at the origin $O$.
OrthocentrePoint where the three altitudes meet at the point $H(5, 0)$.
There are nine such triangles having a perimeter $\le 50$.
Listed and shown in ascending o... | <p>Consider all the triangles having:
</p><ul><li>All their vertices on <strong class="tooltip">lattice points<span class="tooltiptext">Integer coordinates</span></strong>.</li>
<li><strong class="tooltip">Circumcentre<span class="tooltiptext">Centre of the circumscribed circle</span></strong> at the origin $O$.</li>
<... | 2816417.1055 | Saturday, 14th November 2009, 05:00 am | 640 | 85% | hard |
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