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 |
|---|---|---|---|---|---|---|---|---|---|
344 | Silver Dollar Game | One variant of N.G. de Bruijn's silver dollar game can be described as follows:
On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the silver dollar, has any value. Two players take turns making moves. At each turn a player must make either a regular or a special move... | One variant of N.G. de Bruijn's silver dollar game can be described as follows:
On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the silver dollar, has any value. Two players take turns making moves. At each turn a player must make either a regular or a special move... | <p>One variant of N.G. de Bruijn's <strong>silver dollar</strong> game can be described as follows:</p>
<p>On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the <strong>silver dollar</strong>, has any value. Two players take turns making moves. At each turn a player ... | 65579304332 | Saturday, 25th June 2011, 07:00 pm | 374 | 100% | hard |
179 | Consecutive Positive Divisors | Find the number of integers $1 \lt n \lt 10^7$, for which $n$ and $n + 1$ have the same number of positive divisors. For example, $14$ has the positive divisors $1, 2, 7, 14$ while $15$ has $1, 3, 5, 15$. | Find the number of integers $1 \lt n \lt 10^7$, for which $n$ and $n + 1$ have the same number of positive divisors. For example, $14$ has the positive divisors $1, 2, 7, 14$ while $15$ has $1, 3, 5, 15$. | <p>Find the number of integers $1 \lt n \lt 10^7$, for which $n$ and $n + 1$ have the same number of positive divisors. For example, $14$ has the positive divisors $1, 2, 7, 14$ while $15$ has $1, 3, 5, 15$.</p> | 986262 | Saturday, 26th January 2008, 05:00 am | 12240 | 25% | easy |
728 | Circle of Coins | Consider $n$ coins arranged in a circle where each coin shows heads or tails. A move consists of turning over $k$ consecutive coins: tail-head or head-tail. Using a sequence of these moves the objective is to get all the coins showing heads.
Consider the example, shown below, where $n=8$ and $k=3$ and the initial state... | Consider $n$ coins arranged in a circle where each coin shows heads or tails. A move consists of turning over $k$ consecutive coins: tail-head or head-tail. Using a sequence of these moves the objective is to get all the coins showing heads.
Consider the example, shown below, where $n=8$ and $k=3$ and the initial state... | <p>Consider $n$ coins arranged in a circle where each coin shows heads or tails. A move consists of turning over $k$ consecutive coins: tail-head or head-tail. Using a sequence of these moves the objective is to get all the coins showing heads.</p>
<p>Consider the example, shown below, where $n=8$ and $k=3$ and the ini... | 709874991 | Sunday, 4th October 2020, 02:00 am | 305 | 40% | medium |
274 | Divisibility Multipliers | For each integer $p \gt 1$ coprime to $10$ there is a positive divisibility multiplier $m \lt p$ which preserves divisibility by $p$ for the following function on any positive integer, $n$:
$f(n) = (\text{all but the last digit of }n) + (\text{the last digit of }n) \cdot m$.
That is, if $m$ is the divisibility multipli... | For each integer $p \gt 1$ coprime to $10$ there is a positive divisibility multiplier $m \lt p$ which preserves divisibility by $p$ for the following function on any positive integer, $n$:
$f(n) = (\text{all but the last digit of }n) + (\text{the last digit of }n) \cdot m$.
That is, if $m$ is the divisibility multipli... | <p>For each integer $p \gt 1$ coprime to $10$ there is a positive <dfn>divisibility multiplier</dfn> $m \lt p$ which preserves divisibility by $p$ for the following function on any positive integer, $n$:</p>
<p>$f(n) = (\text{all but the last digit of }n) + (\text{the last digit of }n) \cdot m$.</p>
<p>That is, if $m$ ... | 1601912348822 | Friday, 15th January 2010, 01:00 pm | 1568 | 65% | hard |
619 | Square Subsets | For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements is a perfect square.
For example $C(5,10)=3$, since the products of all elements of $\{5, 8, 10\}$, $\{5, 8, 9, 10\}$ and $\{9\}$ are perfect squares, and no other subsets of... | For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements is a perfect square.
For example $C(5,10)=3$, since the products of all elements of $\{5, 8, 10\}$, $\{5, 8, 9, 10\}$ and $\{9\}$ are perfect squares, and no other subsets of... | <p>For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements is a perfect square.</p>
<p>For example $C(5,10)=3$, since the products of all elements of $\{5, 8, 10\}$, $\{5, 8, 9, 10\}$ and $\{9\}$ are perfect squares, and no other ... | 857810883 | Saturday, 27th January 2018, 10:00 pm | 445 | 45% | medium |
911 | Khinchin Exceptions | An irrational number $x$ can be uniquely expressed as a continued fraction $[a_0; a_1,a_2,a_3,\dots]$:
$$
x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}
$$where $a_0$ is an integer and $a_1,a_2,a_3,\dots$ are positive integers.
Define $k_j(x)$ to be the geometric mean of $a_1,a_2,\dots,a_j$. That is, ... | An irrational number $x$ can be uniquely expressed as a continued fraction $[a_0; a_1,a_2,a_3,\dots]$:
$$
x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}
$$where $a_0$ is an integer and $a_1,a_2,a_3,\dots$ are positive integers.
Define $k_j(x)$ to be the geometric mean of $a_1,a_2,\dots,a_j$. That is, ... | <p>
An irrational number $x$ can be uniquely expressed as a <b>continued fraction</b> $[a_0; a_1,a_2,a_3,\dots]$:
$$
x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}
$$where $a_0$ is an integer and $a_1,a_2,a_3,\dots$ are positive integers.
</p>
<p>
Define $k_j(x)$ to be the <b>geometric mean</b> of $a_1,... | 5679.934966 | Sunday, 6th October 2024, 08:00 am | 172 | 50% | medium |
909 | L-expressions I | An L-expression is defined as any one of the following:
a natural number;
the symbol $A$;
the symbol $Z$;
the symbol $S$;
a pair of L-expressions $u, v$, which is written as $u(v)$.
An L-expression can be transformed according to the following rules:
$A(x) \to x + 1$ for any natural number $x$;
$Z(u)(v) \to v$ for ... | An L-expression is defined as any one of the following:
a natural number;
the symbol $A$;
the symbol $Z$;
the symbol $S$;
a pair of L-expressions $u, v$, which is written as $u(v)$.
An L-expression can be transformed according to the following rules:
$A(x) \to x + 1$ for any natural number $x$;
$Z(u)(v) \to v$ for ... | <p>
An <dfn>L-expression</dfn> is defined as any one of the following:</p>
<ul>
<li>a natural number;</li>
<li>the symbol $A$;</li>
<li>the symbol $Z$;</li>
<li>the symbol $S$;</li>
<li>a pair of L-expressions $u, v$, which is written as $u(v)$.</li>
</ul>
<p>
An L-expression can be transformed according to the followi... | 399885292 | Sunday, 29th September 2024, 05:00 am | 150 | 70% | hard |
230 | Fibonacci Words | For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the previous two.
Further, we define $D_{A, B}(n)$ to be the $n$th digit in the first term of $F_{A, B}$ that contains at least $n$ digits.
Example:
Let $A=1415926535... | For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the previous two.
Further, we define $D_{A, B}(n)$ to be the $n$th digit in the first term of $F_{A, B}$ that contains at least $n$ digits.
Example:
Let $A=1415926535... | <p>For any two strings of digits, $A$ and $B$, we define $F_{A, B}$ to be the sequence $(A,B,AB,BAB,ABBAB,\dots)$ in which each term is the concatenation of the previous two.</p>
<p>Further, we define $D_{A, B}(n)$ to be the $n$<sup>th</sup> digit in the first term of $F_{A, B}$ that contains at least $n$ digits.</p>
<... | 850481152593119296 | Saturday, 31st January 2009, 01:00 pm | 3131 | 50% | medium |
354 | Distances in a Bee's Honeycomb | Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length $1$.
One particular cell is occupied by the queen bee.
For a positive real number $L$, let $\text{B}(L)$ count the cells with distance $L$ from the queen bee cell (all distances are measured from centre to centre); you may... | Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length $1$.
One particular cell is occupied by the queen bee.
For a positive real number $L$, let $\text{B}(L)$ count the cells with distance $L$ from the queen bee cell (all distances are measured from centre to centre); you may... | <p>Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length $1$.</p>
<div align="center">
<img alt="p354_bee_honeycomb.png" src="project/images/p354_bee_honeycomb.png"/></div>
<p>
One particular cell is occupied by the queen bee.<br>
For a positive real number $L$, let $\text{B}(L)... | 58065134 | Sunday, 16th October 2011, 07:00 am | 499 | 65% | hard |
627 | Counting Products | Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is
$S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n \le m \}$.
Let $F(m,n)$ be the number of the distinct elements of the set $S$.
For example, $F(9, 2) = 36$ and $F(30,2)=308$.
Find $F(30, 10001) \bmod 1\,000\,000\,0... | Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is
$S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n \le m \}$.
Let $F(m,n)$ be the number of the distinct elements of the set $S$.
For example, $F(9, 2) = 36$ and $F(30,2)=308$.
Find $F(30, 10001) \bmod 1\,000\,000\,0... | <p>Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is<br/>
$S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n \le m \}$.<br/>
Let $F(m,n)$ be the number of the distinct elements of the set $S$.<br/>
For example, $F(9, 2) = 36$ and $F(30,2)=308$.</p>
<p>Find $F(30, 10... | 220196142 | Saturday, 19th May 2018, 10:00 pm | 260 | 60% | hard |
893 | Matchsticks | Define $M(n)$ to be the minimum number of matchsticks needed to represent the number $n$.
A number can be represented in digit form or as an expression involving addition and/or multiplication. Also order of operations must be followed, that is multiplication binding tighter than addition. Any other symbols or operati... | Define $M(n)$ to be the minimum number of matchsticks needed to represent the number $n$.
A number can be represented in digit form or as an expression involving addition and/or multiplication. Also order of operations must be followed, that is multiplication binding tighter than addition. Any other symbols or operati... | <p>
Define $M(n)$ to be the minimum number of matchsticks needed to represent the number $n$.</p>
<p>
A number can be represented in digit form or as an expression involving addition and/or multiplication. Also order of operations must be followed, that is multiplication binding tighter than addition. Any other symbols... | 26688208 | Sunday, 2nd June 2024, 11:00 am | 609 | 15% | easy |
799 | Pentagonal Puzzle | Pentagonal numbers are generated by the formula: $P_n = \tfrac 12n(3n-1)$ giving the sequence:
$$1,5,12,22,35, 51,70,92,\ldots $$
Some pentagonal numbers can be expressed as the sum of two other pentagonal numbers.
For example:
$$P_8 = 92 = 22 + 70 = P_4 + P_7$$
3577 is the smallest pentagonal number that can be ex... | Pentagonal numbers are generated by the formula: $P_n = \tfrac 12n(3n-1)$ giving the sequence:
$$1,5,12,22,35, 51,70,92,\ldots $$
Some pentagonal numbers can be expressed as the sum of two other pentagonal numbers.
For example:
$$P_8 = 92 = 22 + 70 = P_4 + P_7$$
3577 is the smallest pentagonal number that can be ex... | <p>
Pentagonal numbers are generated by the formula: $P_n = \tfrac 12n(3n-1)$ giving the sequence:
</p>
$$1,5,12,22,35, 51,70,92,\ldots $$
<p>
Some pentagonal numbers can be expressed as the sum of two other pentagonal numbers.<br>
For example:
</br></p>
$$P_8 = 92 = 22 + 70 = P_4 + P_7$$
<p>
3577 is the smallest penta... | 1096910149053902 | Sunday, 22nd May 2022, 02:00 am | 248 | 50% | medium |
85 | Counting Rectangles | By counting carefully it can be seen that a rectangular grid measuring $3$ by $2$ contains eighteen rectangles:
Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution. | By counting carefully it can be seen that a rectangular grid measuring $3$ by $2$ contains eighteen rectangles:
Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution. | <p>By counting carefully it can be seen that a rectangular grid measuring $3$ by $2$ contains eighteen rectangles:</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0085.png?1678992052"/></div>
<p>Although there exists no rectangular grid that contains exactly two million rectangles, find the ... | 2772 | Friday, 17th December 2004, 06:00 pm | 27355 | 15% | easy |
609 | $\pi$ Sequences | For every $n \ge 1$ the prime-counting function $\pi(n)$ is equal to the number of primes
not exceeding $n$.
E.g. $\pi(6)=3$ and $\pi(100)=25$.
We say that a sequence of integers $u = (u_0,\cdots,u_m)$ is a $\pi$ sequence if
$u_n \ge 1$ for every $n$
$u_{n+1}= \pi(u_n)$
$u$ has two or more elements
For $u_0=1... | For every $n \ge 1$ the prime-counting function $\pi(n)$ is equal to the number of primes
not exceeding $n$.
E.g. $\pi(6)=3$ and $\pi(100)=25$.
We say that a sequence of integers $u = (u_0,\cdots,u_m)$ is a $\pi$ sequence if
$u_n \ge 1$ for every $n$
$u_{n+1}= \pi(u_n)$
$u$ has two or more elements
For $u_0=1... | <p>
For every $n \ge 1$ the <strong>prime-counting</strong> function $\pi(n)$ is equal to the number of primes
not exceeding $n$.<br/>
E.g. $\pi(6)=3$ and $\pi(100)=25$.
</p>
<p>
We say that a sequence of integers $u = (u_0,\cdots,u_m)$ is a <dfn>$\pi$ sequence</dfn> if
</p><ul>
<li> $u_n \ge 1$ for every $n$
</li><l... | 172023848 | Saturday, 9th September 2017, 04:00 pm | 1033 | 20% | easy |
697 | Randomly Decaying Sequence | Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process:
$X_0 = c$ (with probability 1).
For $n>0$, $X_n = U_n X_{n-1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_m)_{m<n}$.
If we desire th... | Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process:
$X_0 = c$ (with probability 1).
For $n>0$, $X_n = U_n X_{n-1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_m)_{m<n}$.
If we desire th... | <p>Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process:</p>
<ul><li>$X_0 = c$ (with probability 1).</li>
<li>For $n>0$, $X_n = U_n X_{n-1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U... | 4343871.06 | Sunday, 12th January 2020, 04:00 am | 639 | 30% | easy |
462 | Permutation of 3-smooth Numbers | A $3$-smooth number is an integer which has no prime factor larger than $3$. For an integer $N$, we define $S(N)$ as the set of $3$-smooth numbers less than or equal to $N$. For example, $S(20) = \{ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 \}$.
We define $F(N)$ as the number of permutations of $S(N)$ in which each element com... | A $3$-smooth number is an integer which has no prime factor larger than $3$. For an integer $N$, we define $S(N)$ as the set of $3$-smooth numbers less than or equal to $N$. For example, $S(20) = \{ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 \}$.
We define $F(N)$ as the number of permutations of $S(N)$ in which each element com... | <p>
A <strong>$3$-smooth number</strong> is an integer which has no prime factor larger than $3$. For an integer $N$, we define $S(N)$ as the set of <span style="white-space:nowrap;">$3$-smooth</span> numbers less than or equal to $N$. For example, $S(20) = \{ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 \}$.
</p>
<p>
We define $F(... | 5.5350769703e1512 | Saturday, 8th March 2014, 07:00 pm | 360 | 60% | hard |
80 | Square Root Digital Expansion | It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.
The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal digits is $47... | It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.
The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal digits is $47... | <p>It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.</p>
<p>The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal dig... | 40886 | Friday, 8th October 2004, 06:00 pm | 22158 | 20% | easy |
374 | Maximum Integer Partition Product | An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers.
Partitions that differ only in the order of their summands are considered the same.
A partition of $n$ into distinct parts is a partition of $n$ in which every part occurs at most once.
The partitions of $5$ into distinct parts ... | An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers.
Partitions that differ only in the order of their summands are considered the same.
A partition of $n$ into distinct parts is a partition of $n$ in which every part occurs at most once.
The partitions of $5$ into distinct parts ... | <p>An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers.</p>
<p>Partitions that differ only in the order of their summands are considered the same.
A partition of $n$ into <b>distinct parts</b> is a partition of $n$ in which every part occurs at most once.</p>
<p>The partitions of ... | 334420941 | Saturday, 3rd March 2012, 07:00 pm | 757 | 40% | medium |
589 | Poohsticks Marathon | Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one of them wins and they have to go home for tea. The game starts as normal with both dropping a stick simultaneously on the upstream side of a bridge. But rather than the ga... | Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one of them wins and they have to go home for tea. The game starts as normal with both dropping a stick simultaneously on the upstream side of a bridge. But rather than the ga... | <p>
Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one of them wins and they have to go home for tea. The game starts as normal with both dropping a stick simultaneously on the upstream side of a bridge. But rather than th... | 131776959.25 | Sunday, 5th February 2017, 07:00 am | 227 | 95% | hard |
48 | Self Powers | The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.
Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$. | The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.
Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$. | <p>The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.</p>
<p>Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$.</p> | 9110846700 | Friday, 18th July 2003, 06:00 pm | 120934 | 5% | easy |
164 | Three Consecutive Digital Sum Limit | How many $20$ digit numbers $n$ (without any leading zero) exist such that no three consecutive digits of $n$ have a sum greater than $9$? | How many $20$ digit numbers $n$ (without any leading zero) exist such that no three consecutive digits of $n$ have a sum greater than $9$? | <p>How many $20$ digit numbers $n$ (without any leading zero) exist such that no three consecutive digits of $n$ have a sum greater than $9$?</p> | 378158756814587 | Saturday, 20th October 2007, 06:00 am | 6369 | 45% | medium |
378 | Triangle Triples | Let $T(n)$ be the nth triangle number, so $T(n) = \dfrac{n(n + 1)}{2}$.
Let $dT(n)$ be the number of divisors of $T(n)$.
E.g.: $T(7) = 28$ and $dT(7) = 6$.
Let $Tr(n)$ be the number of triples $(i, j, k)$ such that $1 \le i \lt j \lt k \le n$ and $dT(i) \gt dT(j) \gt dT(k)$.
$Tr(20) = 14$, $Tr(100) = 5772$, and $Tr(100... | Let $T(n)$ be the nth triangle number, so $T(n) = \dfrac{n(n + 1)}{2}$.
Let $dT(n)$ be the number of divisors of $T(n)$.
E.g.: $T(7) = 28$ and $dT(7) = 6$.
Let $Tr(n)$ be the number of triples $(i, j, k)$ such that $1 \le i \lt j \lt k \le n$ and $dT(i) \gt dT(j) \gt dT(k)$.
$Tr(20) = 14$, $Tr(100) = 5772$, and $Tr(100... | <p>Let $T(n)$ be the n<sup>th</sup> triangle number, so $T(n) = \dfrac{n(n + 1)}{2}$.</p>
<p>Let $dT(n)$ be the number of divisors of $T(n)$.<br>
E.g.: $T(7) = 28$ and $dT(7) = 6$.</br></p>
<p>Let $Tr(n)$ be the number of triples $(i, j, k)$ such that $1 \le i \lt j \lt k \le n$ and $dT(i) \gt dT(j) \gt dT(k)$.<br/>
$T... | 147534623725724718 | Sunday, 1st April 2012, 07:00 am | 981 | 35% | medium |
796 | A Grand Shuffle | A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for a total of $54$ cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately $29.05361725$ cards so that we ... | A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for a total of $54$ cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately $29.05361725$ cards so that we ... | <p>A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for a total of $54$ cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately $29.05361725$ cards so that ... | 43.20649061 | Saturday, 30th April 2022, 05:00 pm | 219 | 55% | medium |
624 | Two Heads Are Better Than One | An unbiased coin is tossed repeatedly until two consecutive heads are obtained. Suppose these occur on the $(M-1)$th and $M$th toss.
Let $P(n)$ be the probability that $M$ is divisible by $n$. For example, the outcomes HH, HTHH, and THTTHH all count towards $P(2)$, but THH and HTTHH do not.
You are given that $P(2) =\... | An unbiased coin is tossed repeatedly until two consecutive heads are obtained. Suppose these occur on the $(M-1)$th and $M$th toss.
Let $P(n)$ be the probability that $M$ is divisible by $n$. For example, the outcomes HH, HTHH, and THTTHH all count towards $P(2)$, but THH and HTTHH do not.
You are given that $P(2) =\... | <p>
An unbiased coin is tossed repeatedly until two consecutive heads are obtained. Suppose these occur on the $(M-1)$th and $M$th toss.<br/>
Let $P(n)$ be the probability that $M$ is divisible by $n$. For example, the outcomes HH, HTHH, and THTTHH all count towards $P(2)$, but THH and HTTHH do not.</p>
<p>
You are giv... | 984524441 | Saturday, 7th April 2018, 01:00 pm | 703 | 30% | easy |
568 | Reciprocal Games II | 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... | 4228020 | Saturday, 3rd September 2016, 04:00 pm | 320 | 55% | medium |
283 | Integer Sided Triangles with Integral Area/perimeter Ratio | Consider the triangle with sides $6$, $8$, and $10$. It can be seen that the perimeter and the area are both equal to $24$.
So the area/perimeter ratio is equal to $1$.
Consider also the triangle with sides $13$, $14$ and $15$. The perimeter equals $42$ while the area is equal to $84$.
So for this triangle the area/p... | Consider the triangle with sides $6$, $8$, and $10$. It can be seen that the perimeter and the area are both equal to $24$.
So the area/perimeter ratio is equal to $1$.
Consider also the triangle with sides $13$, $14$ and $15$. The perimeter equals $42$ while the area is equal to $84$.
So for this triangle the area/p... | <p>
Consider the triangle with sides $6$, $8$, and $10$. It can be seen that the perimeter and the area are both equal to $24$.
So the area/perimeter ratio is equal to $1$.<br/>
Consider also the triangle with sides $13$, $14$ and $15$. The perimeter equals $42$ while the area is equal to $84$.
So for this triangle t... | 28038042525570324 | Friday, 19th March 2010, 09:00 pm | 711 | 75% | hard |
839 | Beans in Bowls | The sequence $S_n$ is defined by $S_0 = 290797$ and $S_n = S_{n - 1}^2 \bmod 50515093$ for $n > 0$.
There are $N$ bowls indexed $0,1,\dots ,N-1$. Initially there are $S_n$ beans in bowl $n$.
At each step, the smallest index $n$ is found such that bowl $n$ has strictly more beans than bowl $n+1$. Then one bean is moved... | The sequence $S_n$ is defined by $S_0 = 290797$ and $S_n = S_{n - 1}^2 \bmod 50515093$ for $n > 0$.
There are $N$ bowls indexed $0,1,\dots ,N-1$. Initially there are $S_n$ beans in bowl $n$.
At each step, the smallest index $n$ is found such that bowl $n$ has strictly more beans than bowl $n+1$. Then one bean is moved... | <p>
The sequence $S_n$ is defined by $S_0 = 290797$ and $S_n = S_{n - 1}^2 \bmod 50515093$ for $n > 0$.</p>
<p>There are $N$ bowls indexed $0,1,\dots ,N-1$. Initially there are $S_n$ beans in bowl $n$.</p>
<p>
At each step, the smallest index $n$ is found such that bowl $n$ has strictly more beans than bowl $n+1$. T... | 150893234438294408 | Saturday, 15th April 2023, 11:00 pm | 374 | 30% | easy |
259 | Reachable Numbers | A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
Uses the digits $1$ through $9$, in that order and exactly once each.
Any successive digits can be concatenated (for example, using the digits $2$, $3$ and $4$ we obtain the number $234$).
Only the fo... | A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
Uses the digits $1$ through $9$, in that order and exactly once each.
Any successive digits can be concatenated (for example, using the digits $2$, $3$ and $4$ we obtain the number $234$).
Only the fo... | <p>A positive integer will be called <dfn>reachable</dfn> if it can result from an arithmetic expression obeying the following rules:</p>
<ul><li>Uses the digits $1$ through $9$, in that order and exactly once each.</li>
<li>Any successive digits can be concatenated (for example, using the digits $2$, $3$ and $4$ we ob... | 20101196798 | Saturday, 10th October 2009, 01:00 pm | 1680 | 70% | hard |
439 | Sum of Sum of Divisors | Let $d(k)$ be the sum of all divisors of $k$.
We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.
For example, $S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59$.
You are given that $S(10^3) = 563576517282$ and $S(10^5) \bmod 10^9 = 215766508$.
Find $S(10^{11}) \bmod 10^9$. | Let $d(k)$ be the sum of all divisors of $k$.
We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.
For example, $S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59$.
You are given that $S(10^3) = 563576517282$ and $S(10^5) \bmod 10^9 = 215766508$.
Find $S(10^{11}) \bmod 10^9$. | <p>Let $d(k)$ be the sum of all divisors of $k$.<br/>
We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.<br/>
For example, $S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59$.</p>
<p>You are given that $S(10^3) = 563576517282$ and $S(10^5) \bmod 10^9 = 215766508$.<br/>
Find $S... | 968697378 | Sunday, 6th October 2013, 04:00 am | 448 | 100% | hard |
161 | Triominoes | A triomino is a shape consisting of three squares joined via the edges.
There are two basic forms:
If all possible orientations are taken into account there are six:
Any $n$ by $m$ grid for which $n \times m$ is divisible by $3$ can be tiled with triominoes.
If we consider tilings that can be obtained by reflection o... | A triomino is a shape consisting of three squares joined via the edges.
There are two basic forms:
If all possible orientations are taken into account there are six:
Any $n$ by $m$ grid for which $n \times m$ is divisible by $3$ can be tiled with triominoes.
If we consider tilings that can be obtained by reflection o... | <p>A triomino is a shape consisting of three squares joined via the edges.
There are two basic forms:</p>
<p class="center"><img alt="" class="dark_img" src="resources/images/0161_trio1.gif?1678992055"/></p>
<p>If all possible orientations are taken into account there are six:</p>
<p class="center"><img alt="" class="d... | 20574308184277971 | Friday, 21st September 2007, 06:00 pm | 2430 | 70% | hard |
75 | Singular Integer Right Triangles | It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.
$\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$
$\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$
$\pu{\mathbf{30} \mathbf{cm}}$: $(5,12,13)$
$\pu{\mathb... | It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.
$\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$
$\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$
$\pu{\mathbf{30} \mathbf{cm}}$: $(5,12,13)$
$\pu{\mathb... | <p>It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.</p>
<ul style="list-style-type:none;">
<li>$\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$</li>
<li>$\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$</... | 161667 | Friday, 30th July 2004, 06:00 pm | 20294 | 25% | easy |
653 | Frictionless Tube | Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end is sealed. The tube contains $N$ marbles of diameter 20 millimetres at designated starting locations, each one initially moving either westward or eastward with com... | Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end is sealed. The tube contains $N$ marbles of diameter 20 millimetres at designated starting locations, each one initially moving either westward or eastward with com... | <p>Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end is sealed. The tube contains $N$ marbles of diameter 20 millimetres at designated starting locations, each one initially moving either westward or eastward with ... | 1130658687 | Sunday, 27th January 2019, 01:00 am | 347 | 45% | medium |
560 | Coprime Nim | Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins.
Let $L(n, k)$ be the number of losing starting positions for the first playe... | Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins.
Let $L(n, k)$ be the number of losing starting positions for the first playe... | <p>Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is <strong>coprime</strong> with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins.</p>
<p>Let $L(n, k)$ be the number of <strong>losing</st... | 994345168 | Saturday, 14th May 2016, 07:00 pm | 371 | 75% | hard |
897 | Maximal $n$-gon in a region | Let $G(n)$ denote the largest possible area of an $n$-gona polygon with $n$ sides contained in the region $\{(x, y) \in \Bbb R^2: x^4 \leq y \leq 1\}$.
For example, $G(3) = 1$ and $G(5)\approx 1.477309771$.
Find $G(101)$ rounded to nine digits after the decimal point. | Let $G(n)$ denote the largest possible area of an $n$-gona polygon with $n$ sides contained in the region $\{(x, y) \in \Bbb R^2: x^4 \leq y \leq 1\}$.
For example, $G(3) = 1$ and $G(5)\approx 1.477309771$.
Find $G(101)$ rounded to nine digits after the decimal point. | <p>
Let $G(n)$ denote the largest possible area of an <strong class="tooltip">$n$-gon<span class="tooltiptext">a polygon with $n$ sides</span></strong> contained in the region $\{(x, y) \in \Bbb R^2: x^4 \leq y \leq 1\}$.<br/>
For example, $G(3) = 1$ and $G(5)\approx 1.477309771$.<br/>
Find $G(101)$ rounded to nine dig... | 1.599827123 | Saturday, 29th June 2024, 11:00 pm | 405 | 25% | easy |
201 | Subsets with a Unique Sum | For any set $A$ of numbers, let $\operatorname{sum}(A)$ be the sum of the elements of $A$.
Consider the set $B = \{1,3,6,8,10,11\}$. There are $20$ subsets of $B$ containing three elements, and their sums are:
\begin{align}
\operatorname{sum}(\{1,3,6\}) &= 10,\\
\operatorname{sum}(\{1,3,8\}) &= 12,\\
\operatorname{sum... | For any set $A$ of numbers, let $\operatorname{sum}(A)$ be the sum of the elements of $A$.
Consider the set $B = \{1,3,6,8,10,11\}$. There are $20$ subsets of $B$ containing three elements, and their sums are:
\begin{align}
\operatorname{sum}(\{1,3,6\}) &= 10,\\
\operatorname{sum}(\{1,3,8\}) &= 12,\\
\operatorname{sum... | <p>For any set $A$ of numbers, let $\operatorname{sum}(A)$ be the sum of the elements of $A$.<br/>
Consider the set $B = \{1,3,6,8,10,11\}$.<br/> There are $20$ subsets of $B$ containing three elements, and their sums are:</p>
\begin{align}
\operatorname{sum}(\{1,3,6\}) &= 10,\\
\operatorname{sum}(\{1,3,8\}) &= 12,\\
... | 115039000 | Saturday, 5th July 2008, 02:00 pm | 2635 | 65% | hard |
689 | Binary Series | For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$.
For example $d_2(0.25) = 1$, $d_i(0.25) = 0$ for $i \ne 2$.
Let $f(x) = \displaystyle{\sum_{i=1}^{\infty}\frac{d_i(x)}{i^2}}$.
Let $p(a)$ be probability that $f(x) \gt a$, given that $x$ is uniformly d... | For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$.
For example $d_2(0.25) = 1$, $d_i(0.25) = 0$ for $i \ne 2$.
Let $f(x) = \displaystyle{\sum_{i=1}^{\infty}\frac{d_i(x)}{i^2}}$.
Let $p(a)$ be probability that $f(x) \gt a$, given that $x$ is uniformly d... | <p>For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$.<br/>
For example $d_2(0.25) = 1$, $d_i(0.25) = 0$ for $i \ne 2$.</p>
<p>Let $f(x) = \displaystyle{\sum_{i=1}^{\infty}\frac{d_i(x)}{i^2}}$.</p>
<p>Let $p(a)$ be probability that $f(x) \gt a$, given t... | 0.56565454 | Sunday, 17th November 2019, 04:00 am | 245 | 60% | hard |
518 | Prime Triples and Geometric Sequences | Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that:
$a$, $b$ and $c$ are prime numbers.
$a \lt b \lt c \lt n$.
$a+1$, $b+1$, and $c+1$ form a geometric sequence.
For example, $S(100) = 1035$ with the following triples:
$(2, 5, 11)$, $(2, 11, 47)$, $(5, 11, 23)$, $(5, 17, 53)$, $(7, 11, 17)$, $(7, 23, 7... | Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that:
$a$, $b$ and $c$ are prime numbers.
$a \lt b \lt c \lt n$.
$a+1$, $b+1$, and $c+1$ form a geometric sequence.
For example, $S(100) = 1035$ with the following triples:
$(2, 5, 11)$, $(2, 11, 47)$, $(5, 11, 23)$, $(5, 17, 53)$, $(7, 11, 17)$, $(7, 23, 7... | <p>Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that:</p>
<ul style="list-style-type:disc;"><li>$a$, $b$ and $c$ are prime numbers.</li>
<li>$a \lt b \lt c \lt n$.</li>
<li>$a+1$, $b+1$, and $c+1$ form a <strong>geometric sequence</strong>.</li>
</ul><p>For example, $S(100) = 1035$ with the following t... | 100315739184392 | Saturday, 30th May 2015, 04:00 pm | 1764 | 20% | easy |
708 | Twos Are All You Need | A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we define $f(1)=1$.
For example, $90 = 2\times 3\times 3\times 5$, then replacing the primes, $2\times 2\times 2\times 2 = 16$, hence $f(90) = 16$.
Let $\displaystyle S(... | A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we define $f(1)=1$.
For example, $90 = 2\times 3\times 3\times 5$, then replacing the primes, $2\times 2\times 2\times 2 = 16$, hence $f(90) = 16$.
Let $\displaystyle S(... | <p>A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we define $f(1)=1$.</p>
<p>For example, $90 = 2\times 3\times 3\times 5$, then replacing the primes, $2\times 2\times 2\times 2 = 16$, hence $f(90) = 16$.</p>
<p>Let ... | 28874142998632109 | Saturday, 28th March 2020, 01:00 pm | 348 | 50% | medium |
907 | Stacking Cups | An infant's toy consists of $n$ cups, labelled $C_1,\dots,C_n$ in increasing order of size.
The cups may be stacked in various combinations and orientations to form towers. The cups are shaped such that the following means of stacking are possible:
Nesting: $C_k$ may sit snugly inside $C_{k+1}$.
Base-to-base: $C... | An infant's toy consists of $n$ cups, labelled $C_1,\dots,C_n$ in increasing order of size.
The cups may be stacked in various combinations and orientations to form towers. The cups are shaped such that the following means of stacking are possible:
Nesting: $C_k$ may sit snugly inside $C_{k+1}$.
Base-to-base: $C... | <p>
An infant's toy consists of $n$ cups, labelled $C_1,\dots,C_n$ in increasing order of size.
</p>
<img alt="0907_four_cups.png" height="162" src="resources/images/0907_four_cups.png?1723769212"/>
<p>
The cups may be stacked in various combinations and orientations to form towers. The cups are shaped such that the fo... | 196808901 | Saturday, 14th September 2024, 11:00 pm | 257 | 35% | medium |
296 | Angular Bisector and Tangent | Given is an integer sided triangle $ABC$ with $BC \le AC \le AB$.$k$ is the angular bisector of angle $ACB$.$m$ is the tangent at $C$ to the circumscribed circle of $ABC$.$n$ is a line parallel to $m$ through $B$.
The intersection of $n$ and $k$ is called $E$.
How many triangles $ABC$ with a perimeter not exceeding ... | Given is an integer sided triangle $ABC$ with $BC \le AC \le AB$.$k$ is the angular bisector of angle $ACB$.$m$ is the tangent at $C$ to the circumscribed circle of $ABC$.$n$ is a line parallel to $m$ through $B$.
The intersection of $n$ and $k$ is called $E$.
How many triangles $ABC$ with a perimeter not exceeding ... | <p>
Given is an integer sided triangle $ABC$ with $BC \le AC \le AB$.<br/>$k$ is the angular bisector of angle $ACB$.<br/>$m$ is the tangent at $C$ to the circumscribed circle of $ABC$.<br/>$n$ is a line parallel to $m$ through $B$.<br/>
The intersection of $n$ and $k$ is called $E$.
</p>
<div align="center"><img alt="... | 1137208419 | Friday, 11th June 2010, 01:00 pm | 668 | 60% | hard |
424 | Kakuro | The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous internet sites. Other related information can also be currently found at krazydad.com whose author has provided the... | The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous internet sites. Other related information can also be currently found at krazydad.com whose author has provided the... | <div class="center"><img alt="p424_kakuro1.gif" class="dark_img" src="project/images/p424_kakuro1.gif"/></div>
<p>The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous i... | 1059760019628 | Saturday, 20th April 2013, 01:00 pm | 452 | 60% | hard |
119 | Digit Power Sum | The number $512$ is interesting because it is equal to the sum of its digits raised to some power: $5 + 1 + 2 = 8$, and $8^3 = 512$. Another example of a number with this property is $614656 = 28^4$.
We shall define $a_n$ to be the $n$th term of this sequence and insist that a number must contain at least two digits to... | The number $512$ is interesting because it is equal to the sum of its digits raised to some power: $5 + 1 + 2 = 8$, and $8^3 = 512$. Another example of a number with this property is $614656 = 28^4$.
We shall define $a_n$ to be the $n$th term of this sequence and insist that a number must contain at least two digits to... | <p>The number $512$ is interesting because it is equal to the sum of its digits raised to some power: $5 + 1 + 2 = 8$, and $8^3 = 512$. Another example of a number with this property is $614656 = 28^4$.</p>
<p>We shall define $a_n$ to be the $n$th term of this sequence and insist that a number must contain at least two... | 248155780267521 | Friday, 7th April 2006, 06:00 pm | 13664 | 30% | easy |
338 | Cutting Rectangular Grid Paper | A rectangular sheet of grid paper with integer dimensions $w \times h$ is given. Its grid spacing is $1$.
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.
For example, from a sheet with dimensions $9 \times 4$, w... | A rectangular sheet of grid paper with integer dimensions $w \times h$ is given. Its grid spacing is $1$.
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.
For example, from a sheet with dimensions $9 \times 4$, w... | <p>A rectangular sheet of grid paper with integer dimensions $w \times h$ is given. Its grid spacing is $1$.<br/>
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.</p>
<p>For example, from a sheet with dimensions ... | 15614292 | Sunday, 15th May 2011, 01:00 am | 371 | 95% | hard |
874 | Maximal Prime Score | Let $p(t)$ denote the $(t+1)$th prime number. So that $p(0) = 2$, $p(1) = 3$, etc.
We define the prime score of a list of nonnegative integers $[a_1, \dots, a_n]$ as the sum $\sum_{i = 1}^n p(a_i)$.
Let $M(k, n)$ be the maximal prime score among all lists $[a_1, \dots, a_n]$ such that:
$0 \leq a_i < k$ for each $i$;
... | Let $p(t)$ denote the $(t+1)$th prime number. So that $p(0) = 2$, $p(1) = 3$, etc.
We define the prime score of a list of nonnegative integers $[a_1, \dots, a_n]$ as the sum $\sum_{i = 1}^n p(a_i)$.
Let $M(k, n)$ be the maximal prime score among all lists $[a_1, \dots, a_n]$ such that:
$0 \leq a_i < k$ for each $i$;
... | <p>
Let $p(t)$ denote the $(t+1)$th prime number. So that $p(0) = 2$, $p(1) = 3$, etc.<br/>
We define the <dfn>prime score</dfn> of a list of nonnegative integers $[a_1, \dots, a_n]$ as the sum $\sum_{i = 1}^n p(a_i)$.<br/>
Let $M(k, n)$ be the maximal prime score among all lists $[a_1, \dots, a_n]$ such that:</p>
<ul>... | 4992775389 | Sunday, 28th January 2024, 04:00 am | 538 | 15% | easy |
216 | The Primality of $2n^2 - 1$ | Consider numbers $t(n)$ of the form $t(n) = 2n^2 - 1$ with $n \gt 1$.
The first such numbers are $7, 17, 31, 49, 71, 97, 127$ and $161$.
It turns out that only $49 = 7 \cdot 7$ and $161 = 7 \cdot 23$ are not prime.
For $n \le 10000$ there are $2202$ numbers $t(n)$ that are prime.
How many numbers $t(n)$ are prime for $... | Consider numbers $t(n)$ of the form $t(n) = 2n^2 - 1$ with $n \gt 1$.
The first such numbers are $7, 17, 31, 49, 71, 97, 127$ and $161$.
It turns out that only $49 = 7 \cdot 7$ and $161 = 7 \cdot 23$ are not prime.
For $n \le 10000$ there are $2202$ numbers $t(n)$ that are prime.
How many numbers $t(n)$ are prime for $... | <p>Consider numbers $t(n)$ of the form $t(n) = 2n^2 - 1$ with $n \gt 1$.<br/>
The first such numbers are $7, 17, 31, 49, 71, 97, 127$ and $161$.<br/>
It turns out that only $49 = 7 \cdot 7$ and $161 = 7 \cdot 23$ are not prime.<br/>
For $n \le 10000$ there are $2202$ numbers $t(n)$ that are prime.</p>
<p>How many numbe... | 5437849 | Friday, 7th November 2008, 05:00 pm | 4604 | 45% | medium |
850 | Fractions of Powers | Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x \rfloor + \{x\}$, where $\lfloor x \rfloor$ (the floor function) is an integer, and $0\le \{x\} < 1$.
For positive integers $k$ and $n$, define the function
\begin{align}
f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n} \right\}
\en... | Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x \rfloor + \{x\}$, where $\lfloor x \rfloor$ (the floor function) is an integer, and $0\le \{x\} < 1$.
For positive integers $k$ and $n$, define the function
\begin{align}
f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n} \right\}
\en... | <p>Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x \rfloor + \{x\}$, where $\lfloor x \rfloor$ (the floor function) is an integer, and $0\le \{x\} < 1$.</p>
<p>For positive integers $k$ and $n$, define the function
\begin{align}
f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n}... | 878255725 | Sunday, 2nd July 2023, 08:00 am | 134 | 85% | hard |
55 | Lychrel Numbers | If we take $47$, reverse and add, $47 + 74 = 121$, which is palindromic.
Not all numbers produce palindromes so quickly. For example,
\begin{align}
349 + 943 &= 1292\\
1292 + 2921 &= 4213\\
4213 + 3124 &= 7337
\end{align}
That is, $349$ took three iterations to arrive at a palindrome.
Although no one has proved it yet,... | If we take $47$, reverse and add, $47 + 74 = 121$, which is palindromic.
Not all numbers produce palindromes so quickly. For example,
\begin{align}
349 + 943 &= 1292\\
1292 + 2921 &= 4213\\
4213 + 3124 &= 7337
\end{align}
That is, $349$ took three iterations to arrive at a palindrome.
Although no one has proved it yet,... | <p>If we take $47$, reverse and add, $47 + 74 = 121$, which is palindromic.</p>
<p>Not all numbers produce palindromes so quickly. For example,</p>
\begin{align}
349 + 943 &= 1292\\
1292 + 2921 &= 4213\\
4213 + 3124 &= 7337
\end{align}
<p>That is, $349$ took three iterations to arrive at a palindrome.</p>
<p>Although n... | 249 | Friday, 24th October 2003, 06:00 pm | 58786 | 5% | easy |
558 | Irrational Base | Let $r$ be the real root of the equation $x^3 = x^2 + 1$.
Every positive integer can be written as the sum of distinct increasing powers of $r$.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, $3 = r^{-10} +... | Let $r$ be the real root of the equation $x^3 = x^2 + 1$.
Every positive integer can be written as the sum of distinct increasing powers of $r$.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, $3 = r^{-10} +... | <p>Let $r$ be the real root of the equation $x^3 = x^2 + 1$.<br/>
Every positive integer can be written as the sum of distinct increasing powers of $r$.<br/>
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.<br/>
For examp... | 226754889 | Saturday, 30th April 2016, 01:00 pm | 275 | 65% | hard |
842 | Irregular Star Polygons | Given $n$ equally spaced points on a circle, we define an $n$-star polygon as an $n$-gon having those $n$ points as vertices. Two $n$-star polygons differing by a rotation/reflection are considered different.
For example, there are twelve $5$-star polygons shown below.
For an $n$-star polygon $S$, let $I(S)$ be the ... | Given $n$ equally spaced points on a circle, we define an $n$-star polygon as an $n$-gon having those $n$ points as vertices. Two $n$-star polygons differing by a rotation/reflection are considered different.
For example, there are twelve $5$-star polygons shown below.
For an $n$-star polygon $S$, let $I(S)$ be the ... | <p>
Given $n$ equally spaced points on a circle, we define an <dfn>$n$-star polygon</dfn> as an $n$-gon having those $n$ points as vertices. Two $n$-star polygons differing by a rotation/reflection are considered <b>different</b>.</p>
<p>
For example, there are twelve $5$-star polygons shown below.</p>
<img alt="0842_5... | 885226002 | Sunday, 7th May 2023, 08:00 am | 135 | 75% | hard |
775 | Saving Paper | When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually. For example, with 10 cubes of unit edge length, it would take 30 units of paper to wrap them in the arrangement shown below, but 60 units to wrap them separately.
Define $g(n)$ to be the maximum am... | When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually. For example, with 10 cubes of unit edge length, it would take 30 units of paper to wrap them in the arrangement shown below, but 60 units to wrap them separately.
Define $g(n)$ to be the maximum am... | <p>When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually. For example, with 10 cubes of unit edge length, it would take 30 units of paper to wrap them in the arrangement shown below, but 60 units to wrap them separately.</p>
<div style="text-align:center... | 946791106 | Sunday, 5th December 2021, 01:00 am | 260 | 40% | medium |
827 | Pythagorean Triple Occurrence | Define $Q(n)$ to be the smallest number that occurs in exactly $n$ Pythagorean triples $(a,b,c)$ where $a \lt b \lt c$.
For example, $15$ is the smallest number occurring in exactly $5$ Pythagorean triples:
$$(9,12,\mathbf{15})\quad (8,\mathbf{15},17)\quad (\mathbf{15},20,25)\quad (\mathbf{15},36,39)\quad (\mathbf{15}... | Define $Q(n)$ to be the smallest number that occurs in exactly $n$ Pythagorean triples $(a,b,c)$ where $a \lt b \lt c$.
For example, $15$ is the smallest number occurring in exactly $5$ Pythagorean triples:
$$(9,12,\mathbf{15})\quad (8,\mathbf{15},17)\quad (\mathbf{15},20,25)\quad (\mathbf{15},36,39)\quad (\mathbf{15}... | <p>
Define $Q(n)$ to be the smallest number that occurs in exactly $n$ <strong>Pythagorean triples</strong> $(a,b,c)$ where $a \lt b \lt c$.</p>
<p>
For example, $15$ is the smallest number occurring in exactly $5$ Pythagorean triples:
$$(9,12,\mathbf{15})\quad (8,\mathbf{15},17)\quad (\mathbf{15},20,25)\quad (\mathbf{... | 397289979 | Saturday, 28th January 2023, 01:00 pm | 193 | 50% | medium |
696 | Mahjong | The game of Mahjong is played with tiles belonging to $s$ suits. Each tile also has a number in the range $1\ldots n$, and for each suit/number combination there are exactly four indistinguishable tiles with that suit and number. (The real Mahjong game also contains other bonus tiles, but those will not feature in this... | The game of Mahjong is played with tiles belonging to $s$ suits. Each tile also has a number in the range $1\ldots n$, and for each suit/number combination there are exactly four indistinguishable tiles with that suit and number. (The real Mahjong game also contains other bonus tiles, but those will not feature in this... | <p>The game of Mahjong is played with tiles belonging to $s$ <dfn>suits</dfn>. Each tile also has a <dfn>number</dfn> in the range $1\ldots n$, and for each suit/number combination there are exactly four indistinguishable tiles with that suit and number. (The real Mahjong game also contains other bonus tiles, but those... | 436944244 | Sunday, 5th January 2020, 01:00 am | 194 | 100% | hard |
447 | Retractions C | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let... | <p>
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by <br>
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$. </br></p>
<p>
We will call $f_{n,a,b}$ a <i>retraction</i> if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \... | 530553372 | Saturday, 16th November 2013, 10:00 pm | 350 | 95% | hard |
285 | Pythagorean Odds | Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution.
The square root of the sum $(k \cdot a + 1)^2 + (k \cdot b + 1)^2$ is then computed and rounded to the nearest integer. If the result is equal to $k$, he scores $k$ points; otherwis... | Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution.
The square root of the sum $(k \cdot a + 1)^2 + (k \cdot b + 1)^2$ is then computed and rounded to the nearest integer. If the result is equal to $k$, he scores $k$ points; otherwis... | <p>Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution.<br/>
The square root of the sum $(k \cdot a + 1)^2 + (k \cdot b + 1)^2$ is then computed and rounded to the nearest integer. If the result is equal to $k$, he scores $k$ points; ... | 157055.80999 | Saturday, 3rd April 2010, 05:00 am | 1361 | 55% | medium |
800 | Hybrid Integers | An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a hybrid-integer.
For example, $800 = 2^5 5^2$ is a hybrid-integer.
We define $C(n)$ to be the number of hybrid-integers less than or equal to $n$.
You are given $C(800) = 2$ and $C(800^{800}) = 10790$.
Find $C(800800^{800800})$. | An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a hybrid-integer.
For example, $800 = 2^5 5^2$ is a hybrid-integer.
We define $C(n)$ to be the number of hybrid-integers less than or equal to $n$.
You are given $C(800) = 2$ and $C(800^{800}) = 10790$.
Find $C(800800^{800800})$. | <p>
An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a <dfn>hybrid-integer</dfn>.<br/>
For example, $800 = 2^5 5^2$ is a hybrid-integer.
</p>
<p>
We define $C(n)$ to be the number of hybrid-integers less than or equal to $n$.<br/>
You are given $C(800) = 2$ and $C(800^{800}) = 10790$.
</p>
<p>
F... | 1412403576 | Sunday, 29th May 2022, 05:00 am | 2165 | 5% | easy |
100 | Arranged Probability | If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P(\text{BB}) = (15/21) \times (14/20) = 1/2$.
The next such arrangement, for which there is exactly $50\%$ chance of taking tw... | If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P(\text{BB}) = (15/21) \times (14/20) = 1/2$.
The next such arrangement, for which there is exactly $50\%$ chance of taking tw... | <p>If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, $P(\text{BB}) = (15/21) \times (14/20) = 1/2$.</p>
<p>The next such arrangement, for which there is exactly $50\%$ chance of... | 756872327473 | Friday, 15th July 2005, 06:00 pm | 18422 | 30% | easy |
140 | Modified Fibonacci Golden Nuggets | Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k-1} + G_{k-2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$.
For this problem we shall be concerned with values of $x$ for which $A_G(... | Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k-1} + G_{k-2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$.
For this problem we shall be concerned with values of $x$ for which $A_G(... | <p>Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k-1} + G_{k-2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$.</p>
<p>For this problem we shall be concerned with values of $x$ for w... | 5673835352990 | Saturday, 3rd February 2007, 07:00 am | 4928 | 55% | medium |
738 | Counting Ordered Factorisations | Define $d(n,k)$ to be the number of ways to write $n$ as a product of $k$ ordered integers
\[
n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le x_1\le x_2\le\ldots\le x_k
\]
Further define $D(N,K)$ to be the sum of $d(n,k)$ for $1\le n\le N$ and $1\le k\le K$.
You are given that $D(10, 10) = 153$ and $D(100... | Define $d(n,k)$ to be the number of ways to write $n$ as a product of $k$ ordered integers
\[
n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le x_1\le x_2\le\ldots\le x_k
\]
Further define $D(N,K)$ to be the sum of $d(n,k)$ for $1\le n\le N$ and $1\le k\le K$.
You are given that $D(10, 10) = 153$ and $D(100... | <p>Define $d(n,k)$ to be the number of ways to write $n$ as a product of $k$ ordered integers</p>
\[
n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le x_1\le x_2\le\ldots\le x_k
\]
<p>Further define $D(N,K)$ to be the sum of $d(n,k)$ for $1\le n\le N$ and $1\le k\le K$.</p>
<p>You are given that $D(10, 10) ... | 143091030 | Sunday, 13th December 2020, 07:00 am | 300 | 35% | medium |
321 | Swapping Counters | A horizontal row comprising of $2n + 1$ squares has $n$ red counters placed at one end and $n$ blue counters at the other end, being separated by a single empty square in the centre. For example, when $n = 3$.
A counter can move from one square to the next (slide) or can jump over another counter (hop) as long as the ... | A horizontal row comprising of $2n + 1$ squares has $n$ red counters placed at one end and $n$ blue counters at the other end, being separated by a single empty square in the centre. For example, when $n = 3$.
A counter can move from one square to the next (slide) or can jump over another counter (hop) as long as the ... | <p>A horizontal row comprising of $2n + 1$ squares has $n$ red counters placed at one end and $n$ blue counters at the other end, being separated by a single empty square in the centre. For example, when $n = 3$.</p>
<p></p><div align="center"><img alt="0321_swapping_counters_1.gif" src="resources/images/0321_swapping_... | 2470433131948040 | Sunday, 23rd January 2011, 01:00 am | 1909 | 30% | easy |
247 | Squares Under a Hyperbola | Consider the region constrained by $1 \le x$ and $0 \le y \le 1/x$.
Let $S_1$ be the largest square that can fit under the curve.
Let $S_2$ be the largest square that fits in the remaining area, and so on.
Let the index of $S_n$ be the pair $(\text{left}, \text{below})$ indicating the number of squares to the left of... | Consider the region constrained by $1 \le x$ and $0 \le y \le 1/x$.
Let $S_1$ be the largest square that can fit under the curve.
Let $S_2$ be the largest square that fits in the remaining area, and so on.
Let the index of $S_n$ be the pair $(\text{left}, \text{below})$ indicating the number of squares to the left of... | <p>Consider the region constrained by $1 \le x$ and $0 \le y \le 1/x$.
</p><p>
Let $S_1$ be the largest square that can fit under the curve.<br/>
Let $S_2$ be the largest square that fits in the remaining area, and so on. <br/>
Let the <dfn>index</dfn> of $S_n$ be the pair $(\text{left}, \text{below})$ indicating the n... | 782252 | Friday, 29th May 2009, 09:00 pm | 1639 | 65% | hard |
24 | Lexicographic Permutations | A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What is ... | A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What is ... | <p>A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:</p>
<p class="center">012 021 102 ... | 2783915460 | Friday, 16th August 2002, 06:00 pm | 124535 | 5% | easy |
192 | Best Approximations | Let $x$ be a real number.
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form, with $s \le d$, such that any rational number which is closer to $x$ than $\frac r s$ has a denominator larger than $d$:
$|\frac p q -x | < |\frac r s -x| \Rightarrow q > d$
For exam... | Let $x$ be a real number.
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form, with $s \le d$, such that any rational number which is closer to $x$ than $\frac r s$ has a denominator larger than $d$:
$|\frac p q -x | < |\frac r s -x| \Rightarrow q > d$
For exam... | <p>Let $x$ be a real number.<br>
A <b>best approximation</b> to $x$ for the <b>denominator bound</b> $d$ is a rational number $\frac r s $ in<b> reduced form</b>, with $s \le d$, such that any rational number which is closer to $x$ than $\frac r s$ has a denominator larger than $d$:</br></p>
<div class="center"> $|\fr... | 57060635927998347 | Saturday, 3rd May 2008, 05:00 am | 1875 | 75% | hard |
506 | Clock Sequence | Consider the infinite repeating sequence of digits:
1234321234321234321...
Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the $n$-th value is $n$.
The sequence goes as follows:
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ...
Let $v_n$ be the $n$-th v... | Consider the infinite repeating sequence of digits:
1234321234321234321...
Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the $n$-th value is $n$.
The sequence goes as follows:
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ...
Let $v_n$ be the $n$-th v... | <p>Consider the infinite repeating sequence of digits:<br/>
1234321234321234321...</p>
<p>Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the $n$-th value is $n$.</p>
<p>The sequence goes as follows:<br/>
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ..... | 18934502 | Sunday, 8th March 2015, 04:00 am | 994 | 30% | easy |
265 | Binary Circles | $2^N$ binary digits can be placed in a circle so that all the $N$-digit clockwise subsequences are distinct.
For $N=3$, two such circular arrangements are possible, ignoring rotations:
For the first arrangement, the $3$-digit subsequences, in clockwise order, are:$000$, $001$, $010$, $101$, $011$, $111$, $110$ and $10... | $2^N$ binary digits can be placed in a circle so that all the $N$-digit clockwise subsequences are distinct.
For $N=3$, two such circular arrangements are possible, ignoring rotations:
For the first arrangement, the $3$-digit subsequences, in clockwise order, are:$000$, $001$, $010$, $101$, $011$, $111$, $110$ and $10... | <p>$2^N$ binary digits can be placed in a circle so that all the $N$-digit clockwise subsequences are distinct.</p>
<p>For $N=3$, two such circular arrangements are possible, ignoring rotations:</p>
<div align="center"><img alt="0265_BinaryCircles.gif" class="dark_img" src="resources/images/0265_BinaryCircles.gif?16789... | 209110240768 | Saturday, 21st November 2009, 09:00 am | 4565 | 40% | medium |
891 | Ambiguous Clock | A round clock only has three hands: hour, minute, second. All hands look identical and move continuously. Moreover, there is no number or reference mark so that the "upright position" is unknown. The clock functions the same as a normal 12-hour analogue clock.
Despite the inconvenient design, for most time it is possi... | A round clock only has three hands: hour, minute, second. All hands look identical and move continuously. Moreover, there is no number or reference mark so that the "upright position" is unknown. The clock functions the same as a normal 12-hour analogue clock.
Despite the inconvenient design, for most time it is possi... | <p>
A round clock only has three hands: hour, minute, second. All hands look identical and move continuously. Moreover, there is no number or reference mark so that the "upright position" is unknown. The clock functions the same as a normal 12-hour analogue clock.</p>
<p>
Despite the inconvenient design, for most time ... | 1541414 | Sunday, 19th May 2024, 05:00 am | 154 | 65% | hard |
466 | Distinct Terms in a Multiplication Table | Let $P(m,n)$ be the number of distinct terms in an $m\times n$ multiplication table.
For example, a $3\times 4$ multiplication table looks like this:
$\times$ 12341 12342 24683 36912
There are $8$ distinct terms $\{1,2,3,4,6,8,9,12\}$, therefore $P(3,4) = 8$.
You are given that:
$P(64,64) = 1263$,
$P(12,345) = 1998$, a... | Let $P(m,n)$ be the number of distinct terms in an $m\times n$ multiplication table.
For example, a $3\times 4$ multiplication table looks like this:
$\times$ 12341 12342 24683 36912
There are $8$ distinct terms $\{1,2,3,4,6,8,9,12\}$, therefore $P(3,4) = 8$.
You are given that:
$P(64,64) = 1263$,
$P(12,345) = 1998$, a... | <p>Let $P(m,n)$ be the number of <i>distinct</i> terms in an $m\times n$ multiplication table.</p>
<p>For example, a $3\times 4$ multiplication table looks like this:</p>
<p></p><center><table class="p466"><tr><th>$\times$</th> <th>1</th><th>2</th><th>3</th><th>4</th></tr><tr><th>1</th> <td>1</td><td>2</td><td>3</td><t... | 258381958195474745 | Sunday, 6th April 2014, 07:00 am | 360 | 65% | hard |
723 | Pythagorean Quadrilaterals | A pythagorean triangle with catheti $a$ and $b$ and hypotenuse $c$ is characterized by the well-known equation $a^2+b^2=c^2$. However, this can also be formulated differently:
When inscribed into a circle with radius $r$, a triangle with sides $a$, $b$ and $c$ is pythagorean, if and only if $a^2+b^2+c^2=8\, r^2$.
Analo... | A pythagorean triangle with catheti $a$ and $b$ and hypotenuse $c$ is characterized by the well-known equation $a^2+b^2=c^2$. However, this can also be formulated differently:
When inscribed into a circle with radius $r$, a triangle with sides $a$, $b$ and $c$ is pythagorean, if and only if $a^2+b^2+c^2=8\, r^2$.
Analo... | <p>A pythagorean triangle with catheti $a$ and $b$ and hypotenuse $c$ is characterized by the well-known equation $a^2+b^2=c^2$. However, this can also be formulated differently:<br/>
When inscribed into a circle with radius $r$, a triangle with sides $a$, $b$ and $c$ is pythagorean, if and only if $a^2+b^2+c^2=8\, r^2... | 1395793419248 | Sunday, 5th July 2020, 08:00 am | 197 | 65% | hard |
553 | Power Sets of Power Sets | Let $P(n)$ be the set of the first $n$ positive integers $\{1, 2, \dots, n\}$.
Let $Q(n)$ be the set of all the non-empty subsets of $P(n)$.
Let $R(n)$ be the set of all the non-empty subsets of $Q(n)$.
An element $X \in R(n)$ is a non-empty subset of $Q(n)$, so it is itself a set.
From $X$ we can construct a graph as ... | Let $P(n)$ be the set of the first $n$ positive integers $\{1, 2, \dots, n\}$.
Let $Q(n)$ be the set of all the non-empty subsets of $P(n)$.
Let $R(n)$ be the set of all the non-empty subsets of $Q(n)$.
An element $X \in R(n)$ is a non-empty subset of $Q(n)$, so it is itself a set.
From $X$ we can construct a graph as ... | <p>Let $P(n)$ be the set of the first $n$ positive integers $\{1, 2, \dots, n\}$.<br/>
Let $Q(n)$ be the set of all the non-empty subsets of $P(n)$.<br/>
Let $R(n)$ be the set of all the non-empty subsets of $Q(n)$.</p>
<p>An element $X \in R(n)$ is a non-empty subset of $Q(n)$, so it is itself a set.<br/>
From $X$ we ... | 57717170 | Saturday, 26th March 2016, 10:00 pm | 234 | 85% | hard |
577 | Counting Hexagons | An equilateral triangle with integer side length $n \ge 3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram below.
The vertices of these triangles constitute a triangular lattice with $\frac{(n+1)(n+2)} 2$ lattice points.
Let $H(n)$ be the number of all regular hexagons that can be... | An equilateral triangle with integer side length $n \ge 3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram below.
The vertices of these triangles constitute a triangular lattice with $\frac{(n+1)(n+2)} 2$ lattice points.
Let $H(n)$ be the number of all regular hexagons that can be... | <p>An equilateral triangle with integer side length $n \ge 3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram below.<br/>
The vertices of these triangles constitute a triangular lattice with $\frac{(n+1)(n+2)} 2$ lattice points.</p>
<p>Let $H(n)$ be the number of all regular hexag... | 265695031399260211 | Saturday, 12th November 2016, 07:00 pm | 1732 | 25% | easy |
594 | Rhombus Tilings | For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations and reflections are counted as separate tilings.
For example, if $O$ is a regular octagon with edge length 1, then $t(O) = 8$. As it happens, all these 8 tilings are rotations ... | For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations and reflections are counted as separate tilings.
For example, if $O$ is a regular octagon with edge length 1, then $t(O) = 8$. As it happens, all these 8 tilings are rotations ... | <p>
For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations and reflections are counted as separate tilings.
</p>
<p>
For example, if $O$ is a regular octagon with edge length 1, then $t(O) = 8$. As it happens, all these 8 tilings are... | 47067598 | Saturday, 11th March 2017, 10:00 pm | 208 | 85% | hard |
770 | Delphi Flip | A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount.
Each round goes as follows:
A chooses and displays, $x$, a nonnegative real number no larger than the amount of gold that A has.
Either B chooses to TAKE. Then A gives B $x$ grams of gold.
Or B chooses to GIVE. Then B gives A $x$... | A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount.
Each round goes as follows:
A chooses and displays, $x$, a nonnegative real number no larger than the amount of gold that A has.
Either B chooses to TAKE. Then A gives B $x$ grams of gold.
Or B chooses to GIVE. Then B gives A $x$... | <p>
A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount.
Each round goes as follows:
</p>
<ul>
<li>
A chooses and displays, $x$, a nonnegative real number no larger than the amount of gold that A has.</li>
<li>
Either B chooses to TAKE. Then A gives B $x$ grams of gold.</li>
<li>
Or B c... | 127311223 | Sunday, 31st October 2021, 10:00 am | 549 | 30% | easy |
521 | Smallest Prime Factor | Let $\operatorname{smpf}(n)$ be the smallest prime factor of $n$.
$\operatorname{smpf}(91)=7$ because $91=7\times 13$ and $\operatorname{smpf}(45)=3$ because $45=3\times 3\times 5$.
Let $S(n)$ be the sum of $\operatorname{smpf}(i)$ for $2 \le i \le n$.
E.g. $S(100)=1257$.
Find $S(10^{12}) \bmod 10^9$. | Let $\operatorname{smpf}(n)$ be the smallest prime factor of $n$.
$\operatorname{smpf}(91)=7$ because $91=7\times 13$ and $\operatorname{smpf}(45)=3$ because $45=3\times 3\times 5$.
Let $S(n)$ be the sum of $\operatorname{smpf}(i)$ for $2 \le i \le n$.
E.g. $S(100)=1257$.
Find $S(10^{12}) \bmod 10^9$. | <p>
Let $\operatorname{smpf}(n)$ be the smallest prime factor of $n$.<br/>
$\operatorname{smpf}(91)=7$ because $91=7\times 13$ and $\operatorname{smpf}(45)=3$ because $45=3\times 3\times 5$.<br/>
Let $S(n)$ be the sum of $\operatorname{smpf}(i)$ for $2 \le i \le n$.<br/>
E.g. $S(100)=1257$.
</p>
<p>
Find $S(10^{12}) \b... | 44389811 | Sunday, 21st June 2015, 01:00 am | 853 | 50% | medium |
437 | Fibonacci Primitive Roots | When we calculate $8^n$ modulo $11$ for $n=0$ to $9$ we get: $1, 8, 9, 6, 4, 10, 3, 2, 5, 7$.
As we see all possible values from $1$ to $10$ occur. So $8$ is a primitive root of $11$.
But there is more:
If we take a closer look we see:
$1+8=9$
$8+9=17 \equiv 6 \bmod 11$
$9+6=15 \equiv 4 \bmod 11$
$6+4=10$
$4+10=14 \equ... | When we calculate $8^n$ modulo $11$ for $n=0$ to $9$ we get: $1, 8, 9, 6, 4, 10, 3, 2, 5, 7$.
As we see all possible values from $1$ to $10$ occur. So $8$ is a primitive root of $11$.
But there is more:
If we take a closer look we see:
$1+8=9$
$8+9=17 \equiv 6 \bmod 11$
$9+6=15 \equiv 4 \bmod 11$
$6+4=10$
$4+10=14 \equ... | <p>
When we calculate $8^n$ modulo $11$ for $n=0$ to $9$ we get: $1, 8, 9, 6, 4, 10, 3, 2, 5, 7$.<br/>
As we see all possible values from $1$ to $10$ occur. So $8$ is a <strong>primitive root</strong> of $11$.<br/>
But there is more:<br/>
If we take a closer look we see:<br/>
$1+8=9$<br/>
$8+9=17 \equiv 6 \bmod 11$<br/... | 74204709657207 | Saturday, 21st September 2013, 10:00 pm | 903 | 35% | medium |
431 | Square Space Silo | Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but he p... | Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but he p... | <p>Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but h... | 23.386029052 | Sunday, 9th June 2013, 10:00 am | 670 | 40% | medium |
263 | An Engineers' Dream Come True | Consider the number $6$. The divisors of $6$ are: $1,2,3$ and $6$.
Every number from $1$ up to and including $6$ can be written as a sum of distinct divisors of $6$:
$1=1$, $2=2$, $3=1+2$, $4=1+3$, $5=2+3$, $6=6$.
A number $n$ is called a practical number if every number from $1$ up to and including $n$ can be expresse... | Consider the number $6$. The divisors of $6$ are: $1,2,3$ and $6$.
Every number from $1$ up to and including $6$ can be written as a sum of distinct divisors of $6$:
$1=1$, $2=2$, $3=1+2$, $4=1+3$, $5=2+3$, $6=6$.
A number $n$ is called a practical number if every number from $1$ up to and including $n$ can be expresse... | <p>
Consider the number $6$. The divisors of $6$ are: $1,2,3$ and $6$.<br/>
Every number from $1$ up to and including $6$ can be written as a sum of distinct divisors of $6$:<br/>
$1=1$, $2=2$, $3=1+2$, $4=1+3$, $5=2+3$, $6=6$.<br/>
A number $n$ is called a practical number if every number from $1$ up to and including ... | 2039506520 | Saturday, 7th November 2009, 01:00 am | 1177 | 75% | hard |
243 | Resilience | A positive fraction whose numerator is less than its denominator is called a proper fraction.
For any denominator, $d$, there will be $d - 1$ proper fractions; for example, with $d = 12$:$1 / 12, 2 / 12, 3 / 12, 4 / 12, 5 / 12, 6 / 12, 7 / 12, 8 / 12, 9 / 12, 10 / 12, 11 / 12$.
We shall call a fraction that cannot be ... | A positive fraction whose numerator is less than its denominator is called a proper fraction.
For any denominator, $d$, there will be $d - 1$ proper fractions; for example, with $d = 12$:$1 / 12, 2 / 12, 3 / 12, 4 / 12, 5 / 12, 6 / 12, 7 / 12, 8 / 12, 9 / 12, 10 / 12, 11 / 12$.
We shall call a fraction that cannot be ... | <p>A positive fraction whose numerator is less than its denominator is called a proper fraction.<br/>
For any denominator, $d$, there will be $d - 1$ proper fractions; for example, with $d = 12$:<br/>$1 / 12, 2 / 12, 3 / 12, 4 / 12, 5 / 12, 6 / 12, 7 / 12, 8 / 12, 9 / 12, 10 / 12, 11 / 12$.
</p>
<p>We shall call a frac... | 892371480 | Saturday, 2nd May 2009, 10:00 am | 10366 | 35% | medium |
751 | Concatenation Coincidence | A non-decreasing sequence of integers $a_n$ can be generated from any positive real value $\theta$ by the following procedure:
\begin{align}
\begin{split}
b_1 &= \theta \\
b_n &= \left\lfloor b_{n-1} \right\rfloor \left(b_{n-1} - \left\lfloor b_{n-1} \right\rfloor + 1\right)~~~\forall ~ n \geq 2 \\
a_n &= \left\lfloor ... | A non-decreasing sequence of integers $a_n$ can be generated from any positive real value $\theta$ by the following procedure:
\begin{align}
\begin{split}
b_1 &= \theta \\
b_n &= \left\lfloor b_{n-1} \right\rfloor \left(b_{n-1} - \left\lfloor b_{n-1} \right\rfloor + 1\right)~~~\forall ~ n \geq 2 \\
a_n &= \left\lfloor ... | <p>A non-decreasing sequence of integers $a_n$ can be generated from any positive real value $\theta$ by the following procedure:
\begin{align}
\begin{split}
b_1 &= \theta \\
b_n &= \left\lfloor b_{n-1} \right\rfloor \left(b_{n-1} - \left\lfloor b_{n-1} \right\rfloor + 1\right)~~~\forall ~ n \geq 2 \\
a_n &... | 2.223561019313554106173177 | Saturday, 13th March 2021, 10:00 pm | 2543 | 5% | easy |
597 | Torpids | The Torpids are rowing races held annually in Oxford, following some curious rules:
A division consists of $n$ boats (typically 13), placed in order based on past performance.
All boats within a division start at 40 metre intervals along the river, in order with the highest-placed boat starting furthest upstream.
T... | The Torpids are rowing races held annually in Oxford, following some curious rules:
A division consists of $n$ boats (typically 13), placed in order based on past performance.
All boats within a division start at 40 metre intervals along the river, in order with the highest-placed boat starting furthest upstream.
T... | The Torpids are rowing races held annually in Oxford, following some curious rules:
<ul><li>
A division consists of $n$ boats (typically 13), placed in order based on past performance.
</li><li>
All boats within a division start at 40 metre intervals along the river, in order with the highest-placed boat starting furt... | 0.5001817828 | Sunday, 2nd April 2017, 07:00 am | 188 | 100% | hard |
812 | Dynamical Polynomials | A dynamical polynomial is a monicleading coefficient is $1$ polynomial $f(x)$ with integer coefficients such that $f(x)$ divides $f(x^2-2)$.
For example, $f(x) = x^2 - x - 2$ is a dynamical polynomial because $f(x^2-2) = x^4-5x^2+4 = (x^2 + x -2)f(x)$.
Let $S(n)$ be the number of dynamical polynomials of degree $n$.
Fo... | A dynamical polynomial is a monicleading coefficient is $1$ polynomial $f(x)$ with integer coefficients such that $f(x)$ divides $f(x^2-2)$.
For example, $f(x) = x^2 - x - 2$ is a dynamical polynomial because $f(x^2-2) = x^4-5x^2+4 = (x^2 + x -2)f(x)$.
Let $S(n)$ be the number of dynamical polynomials of degree $n$.
Fo... | <p>A <dfn>dynamical polynomial</dfn> is a <strong class="tooltip">monic<span class="tooltiptext">leading coefficient is $1$</span></strong> polynomial $f(x)$ with integer coefficients such that $f(x)$ divides $f(x^2-2)$.</p>
<p>For example, $f(x) = x^2 - x - 2$ is a dynamical polynomial because $f(x^2-2) = x^4-5x^2+4 =... | 986262698 | Saturday, 15th October 2022, 05:00 pm | 141 | 100% | hard |
487 | Sums of Power Sums | Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers.
For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$.
Let $S_k(n)$ be the sum of $f_k(i)$ for $1 \le i \le n$. For example, $S_4(100) = 35375333830$.
What is $\sum (S_{10000}(10^{12}) \bmod p)$ over all ... | Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers.
For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$.
Let $S_k(n)$ be the sum of $f_k(i)$ for $1 \le i \le n$. For example, $S_4(100) = 35375333830$.
What is $\sum (S_{10000}(10^{12}) \bmod p)$ over all ... | <p>Let $f_k(n)$ be the sum of the $k$<sup>th</sup> powers of the first $n$ positive integers.</p>
<p>For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$.</p>
<p>Let $S_k(n)$ be the sum of $f_k(i)$ for $1 \le i \le n$. For example, $S_4(100) = 35375333830$.</p>
<p>What is $\sum (S_{... | 106650212746 | Saturday, 1st November 2014, 10:00 pm | 731 | 40% | medium |
104 | Pandigital Fibonacci Ends | The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.
It turns out that $F_{541}$, which contains $113$ digits, is the first Fibonacci number for which the last nine digits are $1$-$9$ pandigital (contain all the digits $1$ to $9$, but not necessaril... | The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.
It turns out that $F_{541}$, which contains $113$ digits, is the first Fibonacci number for which the last nine digits are $1$-$9$ pandigital (contain all the digits $1$ to $9$, but not necessaril... | <p>The Fibonacci sequence is defined by the recurrence relation:</p>
<blockquote>$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.</blockquote>
<p>It turns out that $F_{541}$, which contains $113$ digits, is the first Fibonacci number for which the last nine digits are $1$-$9$ pandigital (contain all the di... | 329468 | Friday, 9th September 2005, 06:00 pm | 17904 | 25% | easy |
415 | Titanic Sets | A set of lattice points $S$ is called a titanic set if there exists a line passing through exactly two points in $S$.
An example of a titanic set is $S = \{(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)\}$, where the line passing through $(0, 1)$ and $(2, 0)$ does not pass through any other point in $S$.
On the other h... | A set of lattice points $S$ is called a titanic set if there exists a line passing through exactly two points in $S$.
An example of a titanic set is $S = \{(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)\}$, where the line passing through $(0, 1)$ and $(2, 0)$ does not pass through any other point in $S$.
On the other h... | <p>A set of lattice points $S$ is called a <dfn>titanic set</dfn> if there exists a line passing through exactly two points in $S$.</p>
<p>An example of a titanic set is $S = \{(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)\}$, where the line passing through $(0, 1)$ and $(2, 0)$ does not pass through any other point i... | 55859742 | Sunday, 17th February 2013, 10:00 am | 358 | 100% | hard |
328 | Lowest-cost Search | 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, has a cost equal to the number asked and we get one of three possible answers: "Your guess is lower than the hidden number", or
"Yes, that's it!", or
"Your guess is higher ... | 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, has a cost equal to the number asked and we get one of three possible answers: "Your guess is lower than the hidden number", or
"Yes, that's it!", or
"Your guess is higher ... | <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, has a <u>cost equal to the number asked</u> and we get one of three possible answers:<br/></p><ul><li> "Your guess is lower than the hidden number", or</li>
<li> "Yes, tha... | 260511850222 | Saturday, 12th March 2011, 10:00 pm | 500 | 95% | hard |
105 | Special Subset Sums: Testing | 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 $S(B) \gt S(C)$.
For examp... | 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 $S(B) \gt S(C)$.
For examp... | <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... | 73702 | Friday, 23rd September 2005, 06:00 pm | 9108 | 45% | medium |
772 | Balanceable $k$-bounded Partitions | A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$.
A balanceable partition is a partition that can be further divided into two parts of equal sums.
For example, $3 + 2 + 2 + 2 + 2 + 1$ is a balanceable $3$-bounded partition of $12$ since $3 + 2 + 1... | A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$.
A balanceable partition is a partition that can be further divided into two parts of equal sums.
For example, $3 + 2 + 2 + 2 + 2 + 1$ is a balanceable $3$-bounded partition of $12$ since $3 + 2 + 1... | <p>A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$.</p>
<p>A balanceable partition is a partition that can be further divided into two parts of equal sums.</p>
<p>For example, $3 + 2 + 2 + 2 + 2 + 1$ is a balanceable $3$-bounded partition of $12$... | 83985379 | Saturday, 13th November 2021, 04:00 pm | 555 | 20% | easy |
832 | Mex Sequence | In this problem $\oplus$ is used to represent the bitwise exclusive or of two numbers.
Starting with blank paper repeatedly do the following:
Write down the smallest positive integer $a$ which is currently not on the paper;
Find the smallest positive integer $b$ such that neither $b$ nor $(a \oplus b)$ is currently on... | In this problem $\oplus$ is used to represent the bitwise exclusive or of two numbers.
Starting with blank paper repeatedly do the following:
Write down the smallest positive integer $a$ which is currently not on the paper;
Find the smallest positive integer $b$ such that neither $b$ nor $(a \oplus b)$ is currently on... | <p>
In this problem $\oplus$ is used to represent the bitwise <strong>exclusive or</strong> of two numbers.<br>
Starting with blank paper repeatedly do the following:</br></p>
<ol type="1">
<li>Write down the smallest positive integer $a$ which is currently not on the paper;</li>
<li>Find the smallest positive integer ... | 552839586 | Sunday, 5th March 2023, 04:00 am | 337 | 30% | easy |
769 | Binary Quadratic Form II | Consider the following binary quadratic form:
$$
\begin{align}
f(x,y)=x^2+5xy+3y^2
\end{align}
$$
A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$ such that $q = f(x,y)$ and $\gcd(x,y)=1$.
We are interested in primitive representations of perfect squares. For example:
$... | Consider the following binary quadratic form:
$$
\begin{align}
f(x,y)=x^2+5xy+3y^2
\end{align}
$$
A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$ such that $q = f(x,y)$ and $\gcd(x,y)=1$.
We are interested in primitive representations of perfect squares. For example:
$... | <p>Consider the following binary quadratic form:</p>
$$
\begin{align}
f(x,y)=x^2+5xy+3y^2
\end{align}
$$
<p>A positive integer $q$ has a primitive representation if there exist positive integers $x$ and $y$ such that $q = f(x,y)$ and <span style="white-space:nowrap;">$\gcd(x,y)=1$.</span></p>
<p>We are interested in pr... | 14246712611506 | Sunday, 24th October 2021, 07:00 am | 160 | 90% | hard |
444 | The Roundtable Lottery | A group of $p$ people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £$p$, with no two tickets alike. The goal of the game is for... | A group of $p$ people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £$p$, with no two tickets alike. The goal of the game is for... | <p>A group of $p$ people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £$p$, with no two tickets alike. The goal of the game is ... | 1.200856722e263 | Saturday, 9th November 2013, 07:00 pm | 340 | 60% | hard |
608 | Divisor Sums | Let $D(m,n)=\displaystyle\sum_{d\mid m}\sum_{k=1}^n\sigma_0(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_0(n)$ is the number of divisors of $n$.
You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$.
Find $D(200!,10^{12}) \bmod (10^9 + 7)$. | Let $D(m,n)=\displaystyle\sum_{d\mid m}\sum_{k=1}^n\sigma_0(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_0(n)$ is the number of divisors of $n$.
You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$.
Find $D(200!,10^{12}) \bmod (10^9 + 7)$. | <p>Let $D(m,n)=\displaystyle\sum_{d\mid m}\sum_{k=1}^n\sigma_0(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_0(n)$ is the number of divisors of $n$.<br/>
You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$.</p>
<p>Find $D(200!,10^{12}) \bmod (10^9 + 7)$.</p> | 439689828 | Saturday, 17th June 2017, 04:00 pm | 339 | 80% | hard |
32 | Pandigital Products | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.
The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is $1$ throug... | We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.
The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is $1$ throug... | <p>We shall say that an $n$-digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.</p>
<p>The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is ... | 45228 | Friday, 6th December 2002, 06:00 pm | 78180 | 5% | easy |
615 | The Millionth Number with at Least One Million Prime Factors | Consider the natural numbers having at least $5$ prime factors, which don't have to be distinct. Sorting these numbers by size gives a list which starts with:
$32=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$48=2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
$64=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$72=2 \cdot 2 \cdot 2 \cdot 3 \cdot... | Consider the natural numbers having at least $5$ prime factors, which don't have to be distinct. Sorting these numbers by size gives a list which starts with:
$32=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$48=2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
$64=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$72=2 \cdot 2 \cdot 2 \cdot 3 \cdot... | <p>
Consider the natural numbers having at least $5$ prime factors, which don't have to be distinct.<br/> Sorting these numbers by size gives a list which starts with:
</p>
<ul style="list-style:none;"><li>$32=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$</li>
<li>$48=2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$</li>
<li>$64=2 \cdot 2 \cdo... | 108424772 | Sunday, 3rd December 2017, 10:00 am | 651 | 35% | medium |
384 | Rudin-Shapiro Sequence | Define the sequence $a(n)$ as the number of adjacent pairs of ones in the binary expansion of $n$ (possibly overlapping).
E.g.: $a(5) = a(101_2) = 0$, $a(6) = a(110_2) = 1$, $a(7) = a(111_2) = 2$.
Define the sequence $b(n) = (-1)^{a(n)}$.
This sequence is called the Rudin-Shapiro sequence.
Also consider the summatory s... | Define the sequence $a(n)$ as the number of adjacent pairs of ones in the binary expansion of $n$ (possibly overlapping).
E.g.: $a(5) = a(101_2) = 0$, $a(6) = a(110_2) = 1$, $a(7) = a(111_2) = 2$.
Define the sequence $b(n) = (-1)^{a(n)}$.
This sequence is called the Rudin-Shapiro sequence.
Also consider the summatory s... | <p>Define the sequence $a(n)$ as the number of adjacent pairs of ones in the binary expansion of $n$ (possibly overlapping).
<br/>E.g.: $a(5) = a(101_2) = 0$, $a(6) = a(110_2) = 1$, $a(7) = a(111_2) = 2$.</p>
<p>Define the sequence $b(n) = (-1)^{a(n)}$.
<br/>This sequence is called the <strong>Rudin-Shapiro</strong> se... | 3354706415856332783 | Sunday, 13th May 2012, 02:00 am | 366 | 65% | hard |
255 | Rounded Square Roots | We define the rounded-square-root of a positive integer $n$ as the square root of $n$ rounded to the nearest integer.
The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of $n$:
Let $d$ be the number of digits of the number $n$.
If $d$ is odd, set $x_0 = 2 \t... | We define the rounded-square-root of a positive integer $n$ as the square root of $n$ rounded to the nearest integer.
The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of $n$:
Let $d$ be the number of digits of the number $n$.
If $d$ is odd, set $x_0 = 2 \t... | <p>We define the <dfn>rounded-square-root</dfn> of a positive integer $n$ as the square root of $n$ rounded to the nearest integer.</p>
<p>The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of $n$:</p>
<p>Let $d$ be the number of digits of the number $n$.<br... | 4.4474011180 | Friday, 11th September 2009, 09:00 pm | 987 | 75% | hard |
18 | Maximum Path Sum I | By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is $23$.
37 4
2 4 6
8 5 9 3
That is, $3 + 7 + 4 + 9 = 23$.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88... | By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is $23$.
37 4
2 4 6
8 5 9 3
That is, $3 + 7 + 4 + 9 = 23$.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88... | <p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is $23$.</p>
<p class="monospace center"><span class="red"><b>3</b></span><br/><span class="red"><b>7</b></span> 4<br/>
2 <span class="red"><b>4</b></span> 6<br/>
8 5 <span class="red"><... | 1074 | Friday, 31st May 2002, 06:00 pm | 157085 | 5% | easy |
25 | $1000$-digit Fibonacci Number | The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.
Hence the first $12$ terms will be:
\begin{align}
F_1 &= 1\\
F_2 &= 1\\
F_3 &= 2\\
F_4 &= 3\\
F_5 &= 5\\
F_6 &= 8\\
F_7 &= 13\\
F_8 &= 21\\
F_9 &= 34\\
F_{10} &= 55\\
F_{11} &= 89\\
F_{12} &= 144
... | The Fibonacci sequence is defined by the recurrence relation:
$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.
Hence the first $12$ terms will be:
\begin{align}
F_1 &= 1\\
F_2 &= 1\\
F_3 &= 2\\
F_4 &= 3\\
F_5 &= 5\\
F_6 &= 8\\
F_7 &= 13\\
F_8 &= 21\\
F_9 &= 34\\
F_{10} &= 55\\
F_{11} &= 89\\
F_{12} &= 144
... | <p>The Fibonacci sequence is defined by the recurrence relation:</p>
<blockquote>$F_n = F_{n - 1} + F_{n - 2}$, where $F_1 = 1$ and $F_2 = 1$.</blockquote>
<p>Hence the first $12$ terms will be:</p>
\begin{align}
F_1 &= 1\\
F_2 &= 1\\
F_3 &= 2\\
F_4 &= 3\\
F_5 &= 5\\
F_6 &= 8\\
F_7 &= 13\\
F_8 &= 21\\
F_9 &= 34\\
F_{10... | 4782 | Friday, 30th August 2002, 06:00 pm | 167940 | 5% | easy |
356 | Largest Roots of Cubic Polynomials | Let $a_n$ be the largest real root of a polynomial $g(x) = x^3 - 2^n \cdot x^2 + n$.
For example, $a_2 = 3.86619826\cdots$
Find the last eight digits of $\sum \limits_{i = 1}^{30} \lfloor a_i^{987654321} \rfloor$.
Note: $\lfloor a \rfloor$ represents the floor function. | Let $a_n$ be the largest real root of a polynomial $g(x) = x^3 - 2^n \cdot x^2 + n$.
For example, $a_2 = 3.86619826\cdots$
Find the last eight digits of $\sum \limits_{i = 1}^{30} \lfloor a_i^{987654321} \rfloor$.
Note: $\lfloor a \rfloor$ represents the floor function. | <p>
Let $a_n$ be the largest real root of a polynomial $g(x) = x^3 - 2^n \cdot x^2 + n$.<br/>
For example, $a_2 = 3.86619826\cdots$</p>
<p>
Find the last eight digits of $\sum \limits_{i = 1}^{30} \lfloor a_i^{987654321} \rfloor$.</p>
<p>
<u><i>Note</i></u>: $\lfloor a \rfloor$ represents the floor function.</p> | 28010159 | Saturday, 29th October 2011, 01:00 pm | 675 | 60% | hard |
865 | Triplicate Numbers | A triplicate number is a positive integer such that, after repeatedly removing three consecutive identical digits from it, all its digits can be removed.
For example, the integer $122555211$ is a triplicate number:
$$122{\color{red}555}211 \rightarrow 1{\color{red}222}11\rightarrow{\color{red}111}\rightarrow.$$
On the... | A triplicate number is a positive integer such that, after repeatedly removing three consecutive identical digits from it, all its digits can be removed.
For example, the integer $122555211$ is a triplicate number:
$$122{\color{red}555}211 \rightarrow 1{\color{red}222}11\rightarrow{\color{red}111}\rightarrow.$$
On the... | <p>
A <dfn>triplicate number</dfn> is a positive integer such that, after repeatedly removing three consecutive identical digits from it, all its digits can be removed.</p>
<p>
For example, the integer $122555211$ is a triplicate number:
$$122{\color{red}555}211 \rightarrow 1{\color{red}222}11\rightarrow{\color{red}111... | 761181918 | Sunday, 26th November 2023, 01:00 am | 285 | 35% | medium |
379 | Least Common Multiple Count | Let $f(n)$ be the number of couples $(x, y)$ with $x$ and $y$ positive integers, $x \le y$ and the least common multiple of $x$ and $y$ equal to $n$.
Let $g$ be the summatory function of $f$, i.e.:
$g(n) = \sum f(i)$ for $1 \le i \le n$.
You are given that $g(10^6) = 37429395$.
Find $g(10^{12})$. | Let $f(n)$ be the number of couples $(x, y)$ with $x$ and $y$ positive integers, $x \le y$ and the least common multiple of $x$ and $y$ equal to $n$.
Let $g$ be the summatory function of $f$, i.e.:
$g(n) = \sum f(i)$ for $1 \le i \le n$.
You are given that $g(10^6) = 37429395$.
Find $g(10^{12})$. | <p>
Let $f(n)$ be the number of couples $(x, y)$ with $x$ and $y$ positive integers, $x \le y$ and the least common multiple of $x$ and $y$ equal to $n$.
</p>
<p>
Let $g$ be the <strong>summatory function</strong> of $f$, i.e.:
$g(n) = \sum f(i)$ for $1 \le i \le n$.
</p><p>
You are given that $g(10^6) = 37429395$.
</... | 132314136838185 | Sunday, 8th April 2012, 11:00 am | 601 | 70% | hard |
713 | Turán's Water Heating System | Turan has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one in the shed. (Nowadays old fashioned fuses are often replaced with reusable mini circuit breakers, but Turan's system still uses old fashioned fuses.)
For the heating syste... | Turan has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one in the shed. (Nowadays old fashioned fuses are often replaced with reusable mini circuit breakers, but Turan's system still uses old fashioned fuses.)
For the heating syste... | <p>
Turan has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one in the shed. (Nowadays old fashioned fuses are often replaced with reusable mini circuit breakers, but Turan's system still uses old fashioned fuses.)
For the heating s... | 788626351539895 | Sunday, 26th April 2020, 02:00 am | 783 | 20% | easy |
789 | Minimal Pairing Modulo $p$ | Given an odd prime $p$, put the numbers $1,...,p-1$ into $\frac{p-1}{2}$ pairs such that each number appears exactly once. Each pair $(a,b)$ has a cost of $ab \bmod p$. For example, if $p=5$ the pair $(3,4)$ has a cost of $12 \bmod 5 = 2$.
The total cost of a pairing is the sum of the costs of its pairs. We say that su... | Given an odd prime $p$, put the numbers $1,...,p-1$ into $\frac{p-1}{2}$ pairs such that each number appears exactly once. Each pair $(a,b)$ has a cost of $ab \bmod p$. For example, if $p=5$ the pair $(3,4)$ has a cost of $12 \bmod 5 = 2$.
The total cost of a pairing is the sum of the costs of its pairs. We say that su... | <p>Given an odd prime $p$, put the numbers $1,...,p-1$ into $\frac{p-1}{2}$ pairs such that each number appears exactly once. Each pair $(a,b)$ has a cost of $ab \bmod p$. For example, if $p=5$ the pair $(3,4)$ has a cost of $12 \bmod 5 = 2$.</p>
<p>The <i>total cost</i> of a pairing is the sum of the costs of its pair... | 13431419535872807040 | Saturday, 12th March 2022, 07:00 pm | 209 | 50% | medium |
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