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 |
|---|---|---|---|---|---|---|---|---|---|
301 | Nim | Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the three-heap normal-play version of Nim, which works as follows:
At the start of the game there are three heaps of stones.
On each player's turn, the player ... | Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the three-heap normal-play version of Nim, which works as follows:
At the start of the game there are three heaps of stones.
On each player's turn, the player ... | <p><dfn>Nim</dfn> is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.</p>
<p>We'll consider the three-heap normal-play version of Nim, which works as follows:</p>
<ul><li>At the start of the game there are three heaps of stones.</... | 2178309 | Saturday, 11th September 2010, 04:00 pm | 7134 | 15% | easy |
753 | Fermat Equation | Fermat's Last Theorem states that no three positive integers $a$, $b$, $c$ satisfy the equation
$$a^n+b^n=c^n$$
for any integer value of $n$ greater than 2.
For this problem we are only considering the case $n=3$. For certain values of $p$, it is possible to solve the congruence equation:
$$a^3+b^3 \equiv c^3 \pmod{p}... | Fermat's Last Theorem states that no three positive integers $a$, $b$, $c$ satisfy the equation
$$a^n+b^n=c^n$$
for any integer value of $n$ greater than 2.
For this problem we are only considering the case $n=3$. For certain values of $p$, it is possible to solve the congruence equation:
$$a^3+b^3 \equiv c^3 \pmod{p}... | <p>Fermat's Last Theorem states that no three positive integers $a$, $b$, $c$ satisfy the equation
$$a^n+b^n=c^n$$
for any integer value of $n$ greater than 2.</p>
<p>For this problem we are only considering the case $n=3$. For certain values of $p$, it is possible to solve the congruence equation:
$$a^3+b^3 \equiv c^... | 4714126766770661630 | Sunday, 28th March 2021, 05:00 am | 425 | 40% | medium |
623 | Lambda Count | The lambda-calculus is a universal model of computation at the core of functional programming languages. It is based on lambda-terms, a minimal programming language featuring only function definitions, function calls and variables. Lambda-terms are built according to the following rules:
Any variable $x$ (single lette... | The lambda-calculus is a universal model of computation at the core of functional programming languages. It is based on lambda-terms, a minimal programming language featuring only function definitions, function calls and variables. Lambda-terms are built according to the following rules:
Any variable $x$ (single lette... | <p>The <strong>lambda-calculus</strong> is a universal model of computation at the core of functional programming languages. It is based on <strong>lambda-terms</strong>, a minimal programming language featuring only function definitions, function calls and variables. Lambda-terms are built according to the following r... | 3679796 | Sunday, 25th March 2018, 10:00 am | 358 | 45% | medium |
196 | Prime Triplets | Build a triangle from all positive integers in the following way:
1
2 3
4 5 6
7 8 9 1011 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 2829 30 31 32 33 34 35 3637 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66
. . .
Each positive integer has up to eight neighbours in t... | Build a triangle from all positive integers in the following way:
1
2 3
4 5 6
7 8 9 1011 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 2829 30 31 32 33 34 35 3637 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66
. . .
Each positive integer has up to eight neighbours in t... | <p>Build a triangle from all positive integers in the following way:</p>
<p style="font-family:'courier new', monospace;font-weight:bold;margin-left:50px;"> 1<br/>
<span style="color:#FF0000;">2</span> <span style="color:#FF0000;">3</span><br/>
4 <span style="color:#FF0000;">5</span> 6<br/>
<span style="color:#FF... | 322303240771079935 | Friday, 30th May 2008, 06:00 pm | 2904 | 65% | hard |
35 | Circular Primes | The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime.
There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$.
How many circular primes are there below one million? | The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime.
There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$.
How many circular primes are there below one million? | <p>The number, $197$, is called a circular prime because all rotations of the digits: $197$, $971$, and $719$, are themselves prime.</p>
<p>There are thirteen such primes below $100$: $2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79$, and $97$.</p>
<p>How many circular primes are there below one million?</p> | 55 | Friday, 17th January 2003, 06:00 pm | 92333 | 5% | easy |
287 | Quadtree Encoding (a Simple Compression Algorithm) | The quadtree encoding allows us to describe a $2^N \times 2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:
the first bit deals with the complete $2^N \times 2^N$ region;
"0" denotes a split:
the current $2^n \times 2^n$ region is divided into $4$ ... | The quadtree encoding allows us to describe a $2^N \times 2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:
the first bit deals with the complete $2^N \times 2^N$ region;
"0" denotes a split:
the current $2^n \times 2^n$ region is divided into $4$ ... | <p>The quadtree encoding allows us to describe a $2^N \times 2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:
</p><ul><li>the first bit deals with the complete $2^N \times 2^N$ region;</li>
<li>"0" denotes a split:
<br/>the current $2^n \times 2^n... | 313135496 | Saturday, 10th April 2010, 09:00 am | 1605 | 40% | medium |
315 | Digital Root Clocks | Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.... | Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.... | <p></p><div align="center"><img alt="0315_clocks.gif" src="resources/images/0315_clocks.gif?1678992056"/></div>
<p>Sam and Max are asked to transform two digital clocks into two "digital root" clocks.<br/>
A digital root clock is a digital clock that calculates digital roots step by step.</p>
<p>When a clock is fed a n... | 13625242 | Sunday, 19th December 2010, 10:00 am | 3692 | 20% | easy |
548 | Gozinta Chains | A gozinta chainfor $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next.
There are eight gozinta chains for $12$:
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.
Let $g(n)$ be the number of gozinta chains for $n$, so $... | A gozinta chainfor $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next.
There are eight gozinta chains for $12$:
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.
Let $g(n)$ be the number of gozinta chains for $n$, so $... | <p>
A <strong>gozinta chain</strong>for $n$ is a sequence $\{1,a,b,\dots,n\}$ where each element properly divides the next.<br/>
There are eight gozinta chains for $12$:<br/>
$\{1,12\}$, $\{1,2,12\}$, $\{1,2,4,12\}$, $\{1,2,6,12\}$, $\{1,3,12\}$, $\{1,3,6,12\}$, $\{1,4,12\}$ and $\{1,6,12\}$.<br/>
Let $g(n)$ be the nu... | 12144044603581281 | Sunday, 21st February 2016, 07:00 am | 695 | 35% | medium |
334 | Spilling the Beans | In Plato's heaven, there exist an infinite number of bowls in a straight line.
Each bowl either contains some or none of a finite number of beans.
A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls. The game ends when each bowl c... | In Plato's heaven, there exist an infinite number of bowls in a straight line.
Each bowl either contains some or none of a finite number of beans.
A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls. The game ends when each bowl c... | <p>In Plato's heaven, there exist an infinite number of bowls in a straight line.<br/>
Each bowl either contains some or none of a finite number of beans.<br/>
A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls.<br/> The game end... | 150320021261690835 | Saturday, 23rd April 2011, 04:00 pm | 527 | 65% | hard |
183 | Maximum Product of Parts | Let $N$ be a positive integer and let $N$ be split into $k$ equal parts, $r = N/k$, so that $N = r + r + \cdots + r$.
Let $P$ be the product of these parts, $P = r \times r \times \cdots \times r = r^k$.
For example, if $11$ is split into five equal parts, $11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2$, then $P = 2.2^5 = 51.53632$... | Let $N$ be a positive integer and let $N$ be split into $k$ equal parts, $r = N/k$, so that $N = r + r + \cdots + r$.
Let $P$ be the product of these parts, $P = r \times r \times \cdots \times r = r^k$.
For example, if $11$ is split into five equal parts, $11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2$, then $P = 2.2^5 = 51.53632$... | <p>Let $N$ be a positive integer and let $N$ be split into $k$ equal parts, $r = N/k$, so that $N = r + r + \cdots + r$.<br/>
Let $P$ be the product of these parts, $P = r \times r \times \cdots \times r = r^k$.</p>
<p>For example, if $11$ is split into five equal parts, $11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2$, then $P = 2.... | 48861552 | Friday, 22nd February 2008, 05:00 pm | 5136 | 45% | medium |
50 | Consecutive Prime Sum | The prime $41$, can be written as the sum of six consecutive primes:
$$41 = 2 + 3 + 5 + 7 + 11 + 13.$$
This is the longest sum of consecutive primes that adds to a prime below one-hundred.
The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal to $953$.
Which pr... | The prime $41$, can be written as the sum of six consecutive primes:
$$41 = 2 + 3 + 5 + 7 + 11 + 13.$$
This is the longest sum of consecutive primes that adds to a prime below one-hundred.
The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal to $953$.
Which pr... | <p>The prime $41$, can be written as the sum of six consecutive primes:</p>
$$41 = 2 + 3 + 5 + 7 + 11 + 13.$$
<p>This is the longest sum of consecutive primes that adds to a prime below one-hundred.</p>
<p>The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal t... | 997651 | Friday, 15th August 2003, 06:00 pm | 68636 | 5% | easy |
519 | Tricoloured Coin Fountains | An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly two coins in the row below is called a fountain of coins. Let $f(n)$ be the number of possible fountains with $n$ coins. For $4$ coins there are three possible arrangements:
Theref... | An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly two coins in the row below is called a fountain of coins. Let $f(n)$ be the number of possible fountains with $n$ coins. For $4$ coins there are three possible arrangements:
Theref... | <p>An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly two coins in the row below is called a <dfn>fountain</dfn> of coins. Let $f(n)$ be the number of possible fountains with $n$ coins. For $4$ coins there are three possible arrange... | 804739330 | Saturday, 6th June 2015, 07:00 pm | 375 | 50% | medium |
103 | Special Subset Sums: Optimum | Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:
$S(B) \ne S(C)$; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S(B) \gt S(C)$.
If $S(A)$... | Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:
$S(B) \ne S(C)$; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S(B) \gt S(C)$.
If $S(A)$... | <p>Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:</p>
<ol><li>$S(B) \ne S(C)$; that is, sums of subsets cannot be equal.</li>
<li>If $B$ contains more elements than $C$ then $S... | 20313839404245 | Friday, 26th August 2005, 06:00 pm | 9175 | 45% | medium |
688 | Piles of Plates | We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the smallest pile. For example when $n = 10$ and $k = 3$ the piles $2,3,5$ is the best that can be done and so $f(10,3) = 2$. It is impossible to divide 10 into 5 non-empty di... | We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the smallest pile. For example when $n = 10$ and $k = 3$ the piles $2,3,5$ is the best that can be done and so $f(10,3) = 2$. It is impossible to divide 10 into 5 non-empty di... | <p>
We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the smallest pile. For example when $n = 10$ and $k = 3$ the piles $2,3,5$ is the best that can be done and so $f(10,3) = 2$. It is impossible to divide 10 into 5 non-empt... | 110941813 | Sunday, 10th November 2019, 01:00 am | 852 | 20% | easy |
652 | Distinct Values of a Proto-logarithmic Function | Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$.
Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n \ge 2$ has the property that
$f(m_1,n_1)=f(m_2,n_2)$ if
$\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,$ for some integers $a,b,e,f \, \,$ or
$ \, m_1=a^e, n_1=b^e, m... | Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$.
Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n \ge 2$ has the property that
$f(m_1,n_1)=f(m_2,n_2)$ if
$\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,$ for some integers $a,b,e,f \, \,$ or
$ \, m_1=a^e, n_1=b^e, m... | <p>Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$.</p>
<p>Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n \ge 2$ has the property that <br/>
$f(m_1,n_1)=f(m_2,n_2)$ if
</p><ol>
<li>$\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,$ for some integers $a,b,e,f \, \,$ or... | 983924497 | Saturday, 19th January 2019, 10:00 pm | 183 | 70% | hard |
42 | Coded Triangle Numbers | The $n$th term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For exampl... | The $n$th term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For exampl... | <p>The $n$<sup>th</sup> term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$</p>
<p>By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a w... | 162 | Friday, 25th April 2003, 06:00 pm | 80972 | 5% | easy |
554 | Centaurs on a Chess Board | On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an $8 \times 8$ board.
It can be shown that at most $n^2$ non-attacking centaurs can be placed on a board of size $2n \times 2n$.
Let $C(n)$ be the number of ways to plac... | On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an $8 \times 8$ board.
It can be shown that at most $n^2$ non-attacking centaurs can be placed on a board of size $2n \times 2n$.
Let $C(n)$ be the number of ways to plac... | <p>On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an $8 \times 8$ board.</p>
<div align="center"><img alt="0554-centaurs.png" src="resources/images/0554-centaurs.png?1678992053"/></div>
<p>It can be shown that at most ... | 89539872 | Sunday, 3rd April 2016, 01:00 am | 277 | 80% | hard |
367 | Bozo Sort | Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence is sorted and if not swapping randomly two elements. This is repeated until eventually the sequence is sorted.
If we consider all permutations of the first $4$ natural numbers as input the expectat... | Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence is sorted and if not swapping randomly two elements. This is repeated until eventually the sequence is sorted.
If we consider all permutations of the first $4$ natural numbers as input the expectat... | <p>
<strong>Bozo sort</strong>, not to be confused with the slightly less efficient <strong>bogo sort</strong>, consists out of checking if the input sequence is sorted and if not swapping randomly two elements. This is repeated until eventually the sequence is sorted.
</p>
<p>
If we consider all permutations of the fi... | 48271207 | Saturday, 14th January 2012, 10:00 pm | 577 | 55% | medium |
535 | Fractal Sequence | Consider the infinite integer sequence S starting with:
$S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, \dots$
Circle the first occurrence of each integer.
$S = \enclose{circle}1, 1, \enclose{circle}2, 1, \enclose{circle}3, 2, \enclose{circle}4, 1, \enclose{circle}5, 3, \enclose{circle}6, 2, \enclose... | Consider the infinite integer sequence S starting with:
$S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, \dots$
Circle the first occurrence of each integer.
$S = \enclose{circle}1, 1, \enclose{circle}2, 1, \enclose{circle}3, 2, \enclose{circle}4, 1, \enclose{circle}5, 3, \enclose{circle}6, 2, \enclose... | <p>Consider the infinite integer sequence S starting with:<br/>
$S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, \dots$</p>
<p>Circle the first occurrence of each integer.<br/>
$S = \enclose{circle}1, 1, \enclose{circle}2, 1, \enclose{circle}3, 2, \enclose{circle}4, 1, \enclose{circle}5, 3, \enclose{c... | 611778217 | Saturday, 21st November 2015, 04:00 pm | 314 | 60% | hard |
276 | Primitive Triangles | Consider the triangles with integer sides $a$, $b$ and $c$ with $a \le b \le c$.
An integer sided triangle $(a,b,c)$ is called primitive if $\gcd(a, b, c)$$\gcd(a,b,c)=\gcd(a,\gcd(b,c))$$=1$.
How many primitive integer sided triangles exist with a perimeter not exceeding $10\,000\,000$? | Consider the triangles with integer sides $a$, $b$ and $c$ with $a \le b \le c$.
An integer sided triangle $(a,b,c)$ is called primitive if $\gcd(a, b, c)$$\gcd(a,b,c)=\gcd(a,\gcd(b,c))$$=1$.
How many primitive integer sided triangles exist with a perimeter not exceeding $10\,000\,000$? | <p>Consider the triangles with integer sides $a$, $b$ and $c$ with $a \le b \le c$.<br/>
An integer sided triangle $(a,b,c)$ is called primitive if <strong class="tooltip">$\gcd(a, b, c)$<span class="tooltiptext">$\gcd(a,b,c)=\gcd(a,\gcd(b,c))$</span></strong>$=1$. <br/>
How many primitive integer sided triangles exist... | 5777137137739632912 | Friday, 29th January 2010, 09:00 pm | 1209 | 75% | hard |
676 | Matching Digit Sums | Let $d(i,b)$ be the digit sum of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$.
When using different bases, the respective digit sums most of the time deviate from each other, for example $d(9,4)=3 \ne d(9,2)$.
However, for some numbers $i$ there will be a match, like $d(17,4)=d(17,2)=2$.
Let $... | Let $d(i,b)$ be the digit sum of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$.
When using different bases, the respective digit sums most of the time deviate from each other, for example $d(9,4)=3 \ne d(9,2)$.
However, for some numbers $i$ there will be a match, like $d(17,4)=d(17,2)=2$.
Let $... | <p>
Let $d(i,b)$ be the <strong>digit sum</strong> of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$.
When using different bases, the respective digit sums most of the time deviate from each other, for example $d(9,4)=3 \ne d(9,2)$.
</p>
<p>
However, for some numbers $i$ there will be a match, lik... | 3562668074339584 | Saturday, 22nd June 2019, 04:00 pm | 257 | 50% | medium |
833 | Square Triangle Products | Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)} 2$.
A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be found when considering the product of two triangle numbers. For example, $T_2 \cdot T_{24}=3 \cdot 300=30^2$.
Let $S(n)$ be the sum of $c$ for all integers tr... | Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)} 2$.
A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be found when considering the product of two triangle numbers. For example, $T_2 \cdot T_{24}=3 \cdot 300=30^2$.
Let $S(n)$ be the sum of $c$ for all integers tr... | <p>Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)} 2$.<br/>
A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be found when considering the product of two triangle numbers. For example, $T_2 \cdot T_{24}=3 \cdot 300=30^2$.</p>
<p>Let $S(n)$ be the sum of $c$ for ... | 43884302 | Sunday, 12th March 2023, 07:00 am | 178 | 65% | hard |
304 | Primonacci | For any positive integer $n$ the function $\operatorname{next\_prime}(n)$ returns the smallest prime $p$ such that $p \gt n$.
The sequence $a(n)$ is defined by:
$a(1)=\operatorname{next\_prime}(10^{14})$ and $a(n)=\operatorname{next\_prime}(a(n-1))$ for $n \gt 1$.
The Fibonacci sequence $f(n)$ is defined by:
$f(0)=... | For any positive integer $n$ the function $\operatorname{next\_prime}(n)$ returns the smallest prime $p$ such that $p \gt n$.
The sequence $a(n)$ is defined by:
$a(1)=\operatorname{next\_prime}(10^{14})$ and $a(n)=\operatorname{next\_prime}(a(n-1))$ for $n \gt 1$.
The Fibonacci sequence $f(n)$ is defined by:
$f(0)=... | <p>
For any positive integer $n$ the function $\operatorname{next\_prime}(n)$ returns the smallest prime $p$ such that $p \gt n$.
</p>
<p>
The sequence $a(n)$ is defined by:<br/>
$a(1)=\operatorname{next\_prime}(10^{14})$ and $a(n)=\operatorname{next\_prime}(a(n-1))$ for $n \gt 1$.
</p>
<p>
The Fibonacci sequence $f(n)... | 283988410192 | Sunday, 3rd October 2010, 01:00 am | 2437 | 40% | medium |
114 | Counting Block Combinations I | A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one grey square. There are exactly seventeen ways of doing this.
How many ways can a row measuring fifty units in ... | A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one grey square. There are exactly seventeen ways of doing this.
How many ways can a row measuring fifty units in ... | <p>A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one grey square. There are exactly seventeen ways of doing this.</p>
<div class="center">
<img alt="0114.png" src=... | 16475640049 | Friday, 17th February 2006, 06:00 pm | 12093 | 35% | medium |
750 | Optimal Card Stacking | Card Stacking is a game on a computer starting with an array of $N$ cards labelled $1,2,\ldots,N$.
A stack of cards can be moved by dragging horizontally with the mouse to another stack but only when the resulting stack is in sequence. The goal of the game is to combine the cards into a single stack using minimal total... | Card Stacking is a game on a computer starting with an array of $N$ cards labelled $1,2,\ldots,N$.
A stack of cards can be moved by dragging horizontally with the mouse to another stack but only when the resulting stack is in sequence. The goal of the game is to combine the cards into a single stack using minimal total... | <p>
Card Stacking is a game on a computer starting with an array of $N$ cards labelled $1,2,\ldots,N$.
A stack of cards can be moved by dragging horizontally with the mouse to another stack but only when the resulting stack is in sequence. The goal of the game is to combine the cards into a single stack using minimal t... | 160640 | Saturday, 6th March 2021, 07:00 pm | 354 | 35% | medium |
918 | Recursive Sequence Summation | The sequence $a_n$ is defined by $a_1=1$, and then recursively for $n\geq1$:
\begin{align*}
a_{2n} &=2a_n\\
a_{2n+1} &=a_n-3a_{n+1}
\end{align*}
The first ten terms are $1, 2, -5, 4, 17, -10, -17, 8, -47, 34$.
Define $\displaystyle S(N) = \sum_{n=1}^N a_n$. You are given $S(10) = -13$
Find $S(10^{12})$. | The sequence $a_n$ is defined by $a_1=1$, and then recursively for $n\geq1$:
\begin{align*}
a_{2n} &=2a_n\\
a_{2n+1} &=a_n-3a_{n+1}
\end{align*}
The first ten terms are $1, 2, -5, 4, 17, -10, -17, 8, -47, 34$.
Define $\displaystyle S(N) = \sum_{n=1}^N a_n$. You are given $S(10) = -13$
Find $S(10^{12})$. | <p>
The sequence $a_n$ is defined by $a_1=1$, and then recursively for $n\geq1$:
\begin{align*}
a_{2n} &=2a_n\\
a_{2n+1} &=a_n-3a_{n+1}
\end{align*}
The first ten terms are $1, 2, -5, 4, 17, -10, -17, 8, -47, 34$.<br/>
Define $\displaystyle S(N) = \sum_{n=1}^N a_n$. You are given $S(10) = -13$<br/>
Find $S(10^... | -6999033352333308 | Sunday, 24th November 2024, 04:00 am | 931 | 10% | easy |
438 | Integer Part of Polynomial Equation's Solutions | For an $n$-tuple of integers $t = (a_1, \dots, a_n)$, let $(x_1, \dots, x_n)$ be the solutions of the polynomial equation $x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1}x + a_n = 0$.
Consider the following two conditions:
$x_1, \dots, x_n$ are all real.
If $x_1, \dots, x_n$ are sorted, $\lfloor x_i\rfloor = i$ fo... | For an $n$-tuple of integers $t = (a_1, \dots, a_n)$, let $(x_1, \dots, x_n)$ be the solutions of the polynomial equation $x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1}x + a_n = 0$.
Consider the following two conditions:
$x_1, \dots, x_n$ are all real.
If $x_1, \dots, x_n$ are sorted, $\lfloor x_i\rfloor = i$ fo... | <p>
For an $n$-tuple of integers $t = (a_1, \dots, a_n)$, let $(x_1, \dots, x_n)$ be the solutions of the polynomial equation $x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1}x + a_n = 0$.
</p>
<p>
Consider the following two conditions:
</p><ul><li>$x_1, \dots, x_n$ are all real.
</li><li>If $x_1, \dots, x_n$ are sor... | 2046409616809 | Sunday, 29th September 2013, 01:00 am | 302 | 95% | hard |
492 | Exploding Sequence | Define the sequence $a_1, a_2, a_3, \dots$ as:
$a_1 = 1$
$a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$.
Examples:
$a_3 = 2359$
$a_6 = 269221280981320216750489044576319$
$a_6 \bmod 1\,000\,000\,007 = 203064689$
$a_{100} \bmod 1\,000\,000\,007 = 456482974$
Define $B(x,y,n)$ as $\sum (a_n \bmod p)$ for every prime $p$ s... | Define the sequence $a_1, a_2, a_3, \dots$ as:
$a_1 = 1$
$a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$.
Examples:
$a_3 = 2359$
$a_6 = 269221280981320216750489044576319$
$a_6 \bmod 1\,000\,000\,007 = 203064689$
$a_{100} \bmod 1\,000\,000\,007 = 456482974$
Define $B(x,y,n)$ as $\sum (a_n \bmod p)$ for every prime $p$ s... | <p>Define the sequence $a_1, a_2, a_3, \dots$ as:</p>
<ul><li>$a_1 = 1$</li>
<li>$a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$.</li>
</ul><p>
Examples:<br/>
$a_3 = 2359$<br/>
$a_6 = 269221280981320216750489044576319$<br/>
$a_6 \bmod 1\,000\,000\,007 = 203064689$<br/>
$a_{100} \bmod 1\,000\,000\,007 = 456482974$
</p>
<p>... | 242586962923928 | Saturday, 6th December 2014, 01:00 pm | 399 | 60% | hard |
527 | Randomized Binary Search | A secret integer $t$ is selected at random within the range $1 \le t \le n$.
The goal is to guess the value of $t$ by making repeated guesses, via integer $g$. After a guess is made, there are three possible outcomes, in which it will be revealed that either $g \lt t$, $g = t$, or $g \gt t$. Then the process can repea... | A secret integer $t$ is selected at random within the range $1 \le t \le n$.
The goal is to guess the value of $t$ by making repeated guesses, via integer $g$. After a guess is made, there are three possible outcomes, in which it will be revealed that either $g \lt t$, $g = t$, or $g \gt t$. Then the process can repea... | <p>A secret integer $t$ is selected at random within the range $1 \le t \le n$. </p>
<p>The goal is to guess the value of $t$ by making repeated guesses, via integer $g$. After a guess is made, there are three possible outcomes, in which it will be revealed that either $g \lt t$, $g = t$, or $g \gt t$. Then the process... | 11.92412011 | Saturday, 26th September 2015, 04:00 pm | 805 | 30% | easy |
142 | Perfect Square Collection | Find the smallest $x + y + z$ with integers $x \gt y \gt z \gt 0$ such that $x + y$, $x - y$, $x + z$, $x - z$, $y + z$, $y - z$ are all perfect squares. | Find the smallest $x + y + z$ with integers $x \gt y \gt z \gt 0$ such that $x + y$, $x - y$, $x + z$, $x - z$, $y + z$, $y - z$ are all perfect squares. | <p>Find the smallest $x + y + z$ with integers $x \gt y \gt z \gt 0$ such that $x + y$, $x - y$, $x + z$, $x - z$, $y + z$, $y - z$ are all perfect squares.</p> | 1006193 | Saturday, 24th February 2007, 01:00 am | 6495 | 45% | medium |
707 | Lights Out | Consider a $w\times h$ grid. A cell is either ON or OFF. When a cell is selected, that cell and all cells connected to that cell by an edge are toggled on-off, off-on. See the diagram for the 3 cases of selecting a corner cell, an edge cell or central cell in a grid that has all cells on (white).
The goal is to get ... | Consider a $w\times h$ grid. A cell is either ON or OFF. When a cell is selected, that cell and all cells connected to that cell by an edge are toggled on-off, off-on. See the diagram for the 3 cases of selecting a corner cell, an edge cell or central cell in a grid that has all cells on (white).
The goal is to get ... | <p>
Consider a $w\times h$ grid. A cell is either ON or OFF. When a cell is selected, that cell and all cells connected to that cell by an edge are toggled on-off, off-on. See the diagram for the 3 cases of selecting a corner cell, an edge cell or central cell in a grid that has all cells on (white).
</p>
<div class="c... | 652907799 | Sunday, 22nd March 2020, 10:00 am | 254 | 55% | medium |
549 | Divisibility of Factorials | The smallest number $m$ such that $10$ divides $m!$ is $m=5$.
The smallest number $m$ such that $25$ divides $m!$ is $m=10$.
Let $s(n)$ be the smallest number $m$ such that $n$ divides $m!$.
So $s(10)=5$ and $s(25)=10$.
Let $S(n)$ be $\sum s(i)$ for $2 \le i \le n$.
$S(100)=2012$.
Find $S(10^8)$. | The smallest number $m$ such that $10$ divides $m!$ is $m=5$.
The smallest number $m$ such that $25$ divides $m!$ is $m=10$.
Let $s(n)$ be the smallest number $m$ such that $n$ divides $m!$.
So $s(10)=5$ and $s(25)=10$.
Let $S(n)$ be $\sum s(i)$ for $2 \le i \le n$.
$S(100)=2012$.
Find $S(10^8)$. | <p>
The smallest number $m$ such that $10$ divides $m!$ is $m=5$.<br/>
The smallest number $m$ such that $25$ divides $m!$ is $m=10$.<br/>
</p>
<p>
Let $s(n)$ be the smallest number $m$ such that $n$ divides $m!$.<br/>
So $s(10)=5$ and $s(25)=10$.<br/>
Let $S(n)$ be $\sum s(i)$ for $2 \le i \le n$.<br/>
$S(100)=2012$.
... | 476001479068717 | Sunday, 28th February 2016, 10:00 am | 3048 | 15% | easy |
610 | Roman Numerals II | A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined by selecting one of these symbols at random, independently of the other items in the sequence. At each step, the seven letters are equally likely to be selected, with probability 14% ea... | A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined by selecting one of these symbols at random, independently of the other items in the sequence. At each step, the seven letters are equally likely to be selected, with probability 14% ea... | <p>A random generator produces a sequence of symbols drawn from the set {<span style="font-family:'courier new', monospace;">I</span>, <span style="font-family:'courier new', monospace;">V</span>, <span style="font-family:'courier new', monospace;">X</span>, <span style="font-family:'courier new', monospace;">L</span>,... | 319.30207833 | Saturday, 23rd September 2017, 07:00 pm | 657 | 35% | medium |
649 | Low-Prime Chessboard Nim | Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$.
The game may start with any arrangement of $c$ coins in squares on the board. It is possible at any time for more than one coin to occupy the same square on the board at the same time. The coins are dist... | Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$.
The game may start with any arrangement of $c$ coins in squares on the board. It is possible at any time for more than one coin to occupy the same square on the board at the same time. The coins are dist... | Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$.
<p>The game may start with any arrangement of $c$ coins in squares on the board. It is possible at any time for more than one coin to occupy the same square on the board at the same time. The coins are d... | 924668016 | Saturday, 29th December 2018, 01:00 pm | 463 | 30% | easy |
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