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 |
|---|---|---|---|---|---|---|---|---|---|
412 | Gnomon Numbering | For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed.
For example, $L(5, 3)$ looks like this:
We want to number each cell of $L(m, n)$ with consecutive integers $1, 2, 3, \dots$ such that the number in every cell is smaller than the number below it ... | For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed.
For example, $L(5, 3)$ looks like this:
We want to number each cell of $L(m, n)$ with consecutive integers $1, 2, 3, \dots$ such that the number in every cell is smaller than the number below it ... | <p>For integers $m, n$ ($0 \leq n \lt m$), let $L(m, n)$ be an $m \times m$ grid with the top-right $n \times n$ grid removed.</p>
<p>For example, $L(5, 3)$ looks like this:</p>
<p class="center"><img alt="0412_table53.png" src="resources/images/0412_table53.png?1678992053"/></p>
<p>We want to number each cell of $L(m,... | 38788800 | Sunday, 27th January 2013, 01:00 am | 556 | 50% | medium |
258 | A Lagged Fibonacci Sequence | A sequence is defined as:
$g_k = 1$, for $0 \le k \le 1999$
$g_k = g_{k-2000} + g_{k - 1999}$, for $k \ge 2000$.
Find $g_k \bmod 20092010$ for $k = 10^{18}$. | A sequence is defined as:
$g_k = 1$, for $0 \le k \le 1999$
$g_k = g_{k-2000} + g_{k - 1999}$, for $k \ge 2000$.
Find $g_k \bmod 20092010$ for $k = 10^{18}$. | <p>A sequence is defined as:</p>
<p>
</p><ul><li>$g_k = 1$, for $0 \le k \le 1999$</li>
<li>$g_k = g_{k-2000} + g_{k - 1999}$, for $k \ge 2000$.
</li></ul><p>Find $g_k \bmod 20092010$ for $k = 10^{18}$.</p> | 12747994 | Saturday, 3rd October 2009, 09:00 am | 1804 | 70% | hard |
740 | Secret Santa | Secret Santa is a process that allows $n$ people to give each other presents, so that each person gives a single present and receives a single present. At the beginning each of the $n$ people write their name on a slip of paper and put the slip into a hat. Each person takes a random slip from the hat. If the slip has t... | Secret Santa is a process that allows $n$ people to give each other presents, so that each person gives a single present and receives a single present. At the beginning each of the $n$ people write their name on a slip of paper and put the slip into a hat. Each person takes a random slip from the hat. If the slip has t... | <p>
Secret Santa is a process that allows $n$ people to give each other presents, so that each person gives a single present and receives a single present. At the beginning each of the $n$ people write their name on a slip of paper and put the slip into a hat. Each person takes a random slip from the hat. If the slip h... | 0.0189581208 | Saturday, 26th December 2020, 01:00 pm | 228 | 55% | medium |
317 | Firecracker | A firecracker explodes at a height of $\pu{100 m}$ above level ground. It breaks into a large number of very small fragments, which move in every direction; all of them have the same initial velocity of $\pu{20 m/s}$.
We assume that the fragments move without air resistance, in a uniform gravitational field with $g=\... | A firecracker explodes at a height of $\pu{100 m}$ above level ground. It breaks into a large number of very small fragments, which move in every direction; all of them have the same initial velocity of $\pu{20 m/s}$.
We assume that the fragments move without air resistance, in a uniform gravitational field with $g=\... | <p>
A firecracker explodes at a height of $\pu{100 m}$ above level ground. It breaks into a large number of very small fragments, which move in every direction; all of them have the same initial velocity of $\pu{20 m/s}$.
</p>
<p>
We assume that the fragments move without air resistance, in a uniform gravitational fiel... | 1856532.8455 | Saturday, 1st January 2011, 04:00 pm | 2974 | 35% | medium |
252 | Convex Holes | Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
As an example, the image below shows a set ... | Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
As an example, the image below shows a set ... | <p>
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
</p>
<p>
As an example, the image below s... | 104924.0 | Saturday, 27th June 2009, 01:00 pm | 965 | 80% | hard |
868 | Belfry Maths | There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.
The same method can be used to create all permutations of a set of letters. Consider the letters to be permuted initially in order from smallest to largest. At each step swap the largest letter with the letter o... | There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.
The same method can be used to create all permutations of a set of letters. Consider the letters to be permuted initially in order from smallest to largest. At each step swap the largest letter with the letter o... | <p>
There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.</p>
<p>
The same method can be used to create all permutations of a set of letters. Consider the letters to be permuted initially in order from smallest to largest. At each step swap the largest letter with t... | 3832914911887589 | Sunday, 17th December 2023, 10:00 am | 458 | 20% | easy |
601 | Divisibility Streaks | For every positive number $n$ we define the function $\mathop{streak}(n)=k$ as the smallest positive integer $k$ such that $n+k$ is not divisible by $k+1$.
E.g:
$13$ is divisible by $1$
$14$ is divisible by $2$
$15$ is divisible by $3$
$16$ is divisible by $4$
$17$ is NOT divisible by $5$
So $\mathop{streak}(13) = 4$. ... | For every positive number $n$ we define the function $\mathop{streak}(n)=k$ as the smallest positive integer $k$ such that $n+k$ is not divisible by $k+1$.
E.g:
$13$ is divisible by $1$
$14$ is divisible by $2$
$15$ is divisible by $3$
$16$ is divisible by $4$
$17$ is NOT divisible by $5$
So $\mathop{streak}(13) = 4$. ... | <p>
For every positive number $n$ we define the function $\mathop{streak}(n)=k$ as the smallest positive integer $k$ such that $n+k$ is not divisible by $k+1$.<br/>
E.g:<br/>
$13$ is divisible by $1$<br/>
$14$ is divisible by $2$<br/>
$15$ is divisible by $3$<br/>
$16$ is divisible by $4$<br/>
$17$ is NOT divisible by ... | 1617243 | Saturday, 29th April 2017, 07:00 pm | 2284 | 20% | easy |
507 | Shortest Lattice Vector | Let $t_n$ be the tribonacci numbers defined as:
$t_0 = t_1 = 0$;
$t_2 = 1$;
$t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$
and let $r_n = t_n \text{ mod } 10^7$.
For each pair of Vectors $V_n=(v_1,v_2,v_3)$ and $W_n=(w_1,w_2,w_3)$ with $v_1=r_{12n-11}-r_{12n-10}, v_2=r_{12n-9}+r_{12n-8}, v_3=r_{12n-7} \cdot r_{12n... | Let $t_n$ be the tribonacci numbers defined as:
$t_0 = t_1 = 0$;
$t_2 = 1$;
$t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$
and let $r_n = t_n \text{ mod } 10^7$.
For each pair of Vectors $V_n=(v_1,v_2,v_3)$ and $W_n=(w_1,w_2,w_3)$ with $v_1=r_{12n-11}-r_{12n-10}, v_2=r_{12n-9}+r_{12n-8}, v_3=r_{12n-7} \cdot r_{12n... | <p>
Let $t_n$ be the <b>tribonacci numbers</b> defined as:<br/>
$t_0 = t_1 = 0$;<br/>
$t_2 = 1$;<br/>
$t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$<br/>
and let $r_n = t_n \text{ mod } 10^7$.
</p>
<p>
For each pair of Vectors $V_n=(v_1,v_2,v_3)$ and $W_n=(w_1,w_2,w_3)$ with $v_1=r_{12n-11}-r_{12n-10}, v_2=r_{12n-9}... | 316558047002627270 | Sunday, 15th March 2015, 07:00 am | 238 | 90% | hard |
79 | Passcode Derivation | A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
The text file, keylog.txt, contains fifty successful login attempts.
Given that t... | A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
The text file, keylog.txt, contains fifty successful login attempts.
Given that t... | <p>A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.</p>
<p>The text file, <a href="resources/documents/0079_keylog.txt">keylog.txt... | 73162890 | Friday, 17th September 2004, 06:00 pm | 44387 | 5% | easy |
241 | Perfection Quotients | For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. For example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$.
A perfect number, as you probably know, is a number with $\sigma(n) = 2n$.
Let us define the perfection quotient of a positive integer as $p(n) = \dfrac{\sigma(n)}{n}$.
Find the sum of all posit... | For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. For example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$.
A perfect number, as you probably know, is a number with $\sigma(n) = 2n$.
Let us define the perfection quotient of a positive integer as $p(n) = \dfrac{\sigma(n)}{n}$.
Find the sum of all posit... | <p>For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. For example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$.</p>
<p>A perfect number, as you probably know, is a number with $\sigma(n) = 2n$.</p>
<p>Let us define the <b>perfection quotient</b> of a positive integer as $p(n) = \dfrac{\sigma(n)}{n}$.</... | 482316491800641154 | Saturday, 18th April 2009, 02:00 am | 1061 | 80% | hard |
377 | Sum of Digits - Experience #13 | There are $16$ positive integers that do not have a zero in their digits and that have a digital sum equal to $5$, namely:
$5$, $14$, $23$, $32$, $41$, $113$, $122$, $131$, $212$, $221$, $311$, $1112$, $1121$, $1211$, $2111$ and $11111$.
Their sum is $17891$.
Let $f(n)$ be the sum of all positive integers that do no... | There are $16$ positive integers that do not have a zero in their digits and that have a digital sum equal to $5$, namely:
$5$, $14$, $23$, $32$, $41$, $113$, $122$, $131$, $212$, $221$, $311$, $1112$, $1121$, $1211$, $2111$ and $11111$.
Their sum is $17891$.
Let $f(n)$ be the sum of all positive integers that do no... | <p>
There are $16$ positive integers that do not have a zero in their digits and that have a digital sum equal to $5$, namely: <br/>
$5$, $14$, $23$, $32$, $41$, $113$, $122$, $131$, $212$, $221$, $311$, $1112$, $1121$, $1211$, $2111$ and $11111$.<br/>
Their sum is $17891$.
</p>
<p>
Let $f(n)$ be the sum of all positiv... | 732385277 | Sunday, 25th March 2012, 05:00 am | 841 | 45% | medium |
215 | Crack-free Walls | Consider the problem of building a wall out of $2 \times 1$ and $3 \times 1$ bricks ($\text{horizontal} \times \text{vertical}$ dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a "running crack".
For example, the following $9 \... | Consider the problem of building a wall out of $2 \times 1$ and $3 \times 1$ bricks ($\text{horizontal} \times \text{vertical}$ dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a "running crack".
For example, the following $9 \... | <p>Consider the problem of building a wall out of $2 \times 1$ and $3 \times 1$ bricks ($\text{horizontal} \times \text{vertical}$ dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a "running crack".</p>
<p>For example, the foll... | 806844323190414 | Friday, 31st October 2008, 01:00 pm | 4037 | 50% | medium |
550 | Divisor Game | Two players are playing a game, alternating turns. There are $k$ piles of stones.
On each turn, a player has to choose a pile and replace it with two piles of stones under the following two conditions:
Both new piles must have a number of stones more than one and less than the number of stones of the original pile.
... | Two players are playing a game, alternating turns. There are $k$ piles of stones.
On each turn, a player has to choose a pile and replace it with two piles of stones under the following two conditions:
Both new piles must have a number of stones more than one and less than the number of stones of the original pile.
... | <p>
Two players are playing a game, alternating turns. There are $k$ piles of stones.
On each turn, a player has to choose a pile and replace it with two piles of stones under the following two conditions:
</p>
<ul><li> Both new piles must have a number of stones more than one and less than the number of stones of the ... | 328104836 | Saturday, 5th March 2016, 01:00 pm | 392 | 60% | hard |
419 | Look and Say Sequence | The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.
It helps to do this out loud:
1 is 'one one' → 11
11 is 'two ones' → 21
21 is 'one two and one one' → ... | The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.
It helps to do this out loud:
1 is 'one one' → 11
11 is 'two ones' → 21
21 is 'one two and one one' → ... | <p>
The <strong>look and say</strong> sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...<br/>
The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.<br/>
It helps to do this out loud:<br/>
1 is 'one one' → 11<br/>
11 is 'two ... | 998567458,1046245404,43363922 | Saturday, 16th March 2013, 10:00 pm | 543 | 50% | medium |
809 | Rational Recurrence Relation | The following is a function defined for all positive rational values of $x$.
$$ f(x)=\begin{cases} x &x\text{ is integral}\\
f(\frac 1{1-x}) &x \lt 1\\
f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) &\text{otherwise}\end{cases} $$
For example, $f(3/2)=3$, $f(1/6) = 65533$ and $f(13/10) = 7625597484985$.
... | The following is a function defined for all positive rational values of $x$.
$$ f(x)=\begin{cases} x &x\text{ is integral}\\
f(\frac 1{1-x}) &x \lt 1\\
f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) &\text{otherwise}\end{cases} $$
For example, $f(3/2)=3$, $f(1/6) = 65533$ and $f(13/10) = 7625597484985$.
... | <p>
The following is a function defined for all positive rational values of $x$.
</p>
$$ f(x)=\begin{cases} x &x\text{ is integral}\\
f(\frac 1{1-x}) &x \lt 1\\
f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) &\text{otherwise}\end{cases} $$
<p>
For example, $f(3/2)=3$, $f(1/6) = 65533$ and $f(13/10) = 7625597... | 75353432948733 | Sunday, 25th September 2022, 08:00 am | 323 | 35% | medium |
271 | Modular Cubes, Part 1 | For a positive number $n$, define $S(n)$ as the sum of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$.
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.
Thus, $S(91)=9+16+22+29+53+74+79+81=363$.
Find $S(13082761331670030)$. | For a positive number $n$, define $S(n)$ as the sum of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$.
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.
Thus, $S(91)=9+16+22+29+53+74+79+81=363$.
Find $S(13082761331670030)$. | <p>
For a positive number $n$, define $S(n)$ as the sum of the integers $x$, for which $1 \lt x \lt n$ and<br/>$x^3 \equiv 1 \bmod n$.
</p>
<p>
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.<br/>
Thus, $S(91)=9+16+22+29+53+74+79+81=363$.</p>
<p>
Find $S(13082761331670030)$.... | 4617456485273129588 | Saturday, 2nd January 2010, 05:00 am | 2654 | 60% | hard |
14 | Longest Collatz Sequence | The following iterative sequence is defined for the set of positive integers:
$n \to n/2$ ($n$ is even)
$n \to 3n + 1$ ($n$ is odd)
Using the rule above and starting with $13$, we generate the following sequence:
$$13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1.$$
It can be seen that this sequence (start... | The following iterative sequence is defined for the set of positive integers:
$n \to n/2$ ($n$ is even)
$n \to 3n + 1$ ($n$ is odd)
Using the rule above and starting with $13$, we generate the following sequence:
$$13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1.$$
It can be seen that this sequence (start... | <p>The following iterative sequence is defined for the set of positive integers:</p>
<ul style="list-style-type:none;">
<li>$n \to n/2$ ($n$ is even)</li>
<li>$n \to 3n + 1$ ($n$ is odd)</li></ul>
<p>Using the rule above and starting with $13$, we generate the following sequence:
$$13 \to 40 \to 20 \to 10 \to 5 \to 16 ... | 837799 | Friday, 5th April 2002, 06:00 pm | 244361 | 5% | easy |
493 | Under the Rainbow | $70$ coloured balls are placed in an urn, $10$ for each of the seven rainbow colours.
What is the expected number of distinct colours in $20$ randomly picked balls?
Give your answer with nine digits after the decimal point (a.bcdefghij). | $70$ coloured balls are placed in an urn, $10$ for each of the seven rainbow colours.
What is the expected number of distinct colours in $20$ randomly picked balls?
Give your answer with nine digits after the decimal point (a.bcdefghij). | <p>$70$ coloured balls are placed in an urn, $10$ for each of the seven rainbow colours.</p>
<p>What is the expected number of distinct colours in $20$ randomly picked balls?</p>
<p>Give your answer with nine digits after the decimal point (a.bcdefghij).</p> | 6.818741802 | Saturday, 13th December 2014, 04:00 pm | 5893 | 10% | easy |
162 | Hexadecimal Numbers | In the hexadecimal number system numbers are represented using $16$ different digits:
$$0,1,2,3,4,5,6,7,8,9,\mathrm A,\mathrm B,\mathrm C,\mathrm D,\mathrm E,\mathrm F.$$
The hexadecimal number $\mathrm{AF}$ when written in the decimal number system equals $10 \times 16 + 15 = 175$.
In the $3$-digit hexadecimal numbers... | In the hexadecimal number system numbers are represented using $16$ different digits:
$$0,1,2,3,4,5,6,7,8,9,\mathrm A,\mathrm B,\mathrm C,\mathrm D,\mathrm E,\mathrm F.$$
The hexadecimal number $\mathrm{AF}$ when written in the decimal number system equals $10 \times 16 + 15 = 175$.
In the $3$-digit hexadecimal numbers... | <p>In the hexadecimal number system numbers are represented using $16$ different digits:
$$0,1,2,3,4,5,6,7,8,9,\mathrm A,\mathrm B,\mathrm C,\mathrm D,\mathrm E,\mathrm F.$$</p>
<p>The hexadecimal number $\mathrm{AF}$ when written in the decimal number system equals $10 \times 16 + 15 = 175$.</p>
<p>In the $3$-digit he... | 3D58725572C62302 | Friday, 5th October 2007, 10:00 pm | 5910 | 45% | medium |
393 | Migrating Ants | An $n \times n$ grid of squares contains $n^2$ ants, one ant per square.
All ants decide to move simultaneously to an adjacent square (usually $4$ possibilities, except for ants on the edge of the grid or at the corners).
We define $f(n)$ to be the number of ways this can happen without any ants ending on the same squa... | An $n \times n$ grid of squares contains $n^2$ ants, one ant per square.
All ants decide to move simultaneously to an adjacent square (usually $4$ possibilities, except for ants on the edge of the grid or at the corners).
We define $f(n)$ to be the number of ways this can happen without any ants ending on the same squa... | <p>
An $n \times n$ grid of squares contains $n^2$ ants, one ant per square.<br/>
All ants decide to move simultaneously to an adjacent square (usually $4$ possibilities, except for ants on the edge of the grid or at the corners).<br/>
We define $f(n)$ to be the number of ways this can happen without any ants ending on... | 112398351350823112 | Saturday, 8th September 2012, 05:00 pm | 852 | 50% | medium |
217 | Balanced Numbers | A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2 \rceil$ digits, where $\lceil x \rceil$, pronounced ceiling of $x$, is the smallest integer $\ge x$, thus $\lceil \pi \rceil = 4$ and $\lceil 5 \rceil = 5$.
So, for examp... | A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2 \rceil$ digits, where $\lceil x \rceil$, pronounced ceiling of $x$, is the smallest integer $\ge x$, thus $\lceil \pi \rceil = 4$ and $\lceil 5 \rceil = 5$.
So, for examp... | <p>
A positive integer with $k$ (decimal) digits is called balanced if its first $\lceil k/2 \rceil$ digits sum to the same value as its last $\lceil k/2 \rceil$ digits, where $\lceil x \rceil$, pronounced <i>ceiling</i> of $x$, is the smallest integer $\ge x$, thus $\lceil \pi \rceil = 4$ and $\lceil 5 \rceil = 5$.</p... | 6273134 | Friday, 14th November 2008, 09:00 pm | 1691 | 70% | hard |
208 | Robot Walks | A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.
One of $70932$ possible closed paths of $25$ arcs starting northward is
Given that the robot starts facing North, how many journeys of $70$ arcs in le... | A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.
One of $70932$ possible closed paths of $25$ arcs starting northward is
Given that the robot starts facing North, how many journeys of $70$ arcs in le... | <p>A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.</p>
<p>One of $70932$ possible closed paths of $25$ arcs starting northward is</p>
<div class="center">
<img alt="" class="dark_img" src="resources/i... | 331951449665644800 | Friday, 12th September 2008, 02:00 pm | 1969 | 70% | hard |
607 | Marsh Crossing | Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The marsh... | Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The marsh... | <p>
Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The m... | 13.1265108586 | Saturday, 10th June 2017, 01:00 pm | 1869 | 20% | easy |
686 | Powers of Two | $2^7=128$ is the first power of two whose leading digits are "12".
The next power of two whose leading digits are "12" is $2^{80}$.
Define $p(L, n)$ to be the $n$th-smallest value of $j$ such that the base 10 representation of $2^j$ begins with the digits of $L$.
So $p(12, 1) = 7$ and $p(12, 2) = 80$.
You are also give... | $2^7=128$ is the first power of two whose leading digits are "12".
The next power of two whose leading digits are "12" is $2^{80}$.
Define $p(L, n)$ to be the $n$th-smallest value of $j$ such that the base 10 representation of $2^j$ begins with the digits of $L$.
So $p(12, 1) = 7$ and $p(12, 2) = 80$.
You are also give... | <p>$2^7=128$ is the first power of two whose leading digits are "12".<br>
The next power of two whose leading digits are "12" is $2^{80}$.</br></p>
<p>Define $p(L, n)$ to be the $n$th-smallest value of $j$ such that the base 10 representation of $2^j$ begins with the digits of $L$.<br/>
So $p(12, 1) = 7$ and $p(12, 2) ... | 193060223 | Saturday, 26th October 2019, 07:00 pm | 3914 | 5% | easy |
113 | Non-bouncy Numbers | Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.
We shall call a positive integer that is neither increasing nor decrea... | Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.
We shall call a positive integer that is neither increasing nor decrea... | <p>Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.</p>
<p>Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.</p>
<p>We shall call a positive integer that is neither incr... | 51161058134250 | Friday, 10th February 2006, 06:00 pm | 12285 | 30% | easy |
508 | Integers in Base $i-1$ | Consider the Gaussian integer $i-1$. A base $i-1$ representation of a Gaussian integer $a+bi$ is a finite sequence of digits $d_{n - 1}d_{n - 2}\cdots d_1 d_0$ such that:
$a+bi = d_{n - 1}(i - 1)^{n - 1} + d_{n - 2}(i - 1)^{n - 2} + \cdots + d_1(i - 1) + d_0$
Each $d_k$ is in $\{0,1\}$
There are no leading zeroes, i.e.... | Consider the Gaussian integer $i-1$. A base $i-1$ representation of a Gaussian integer $a+bi$ is a finite sequence of digits $d_{n - 1}d_{n - 2}\cdots d_1 d_0$ such that:
$a+bi = d_{n - 1}(i - 1)^{n - 1} + d_{n - 2}(i - 1)^{n - 2} + \cdots + d_1(i - 1) + d_0$
Each $d_k$ is in $\{0,1\}$
There are no leading zeroes, i.e.... | <p>Consider the Gaussian integer $i-1$. A <strong>base $i-1$ representation</strong> of a Gaussian integer $a+bi$ is a finite sequence of digits $d_{n - 1}d_{n - 2}\cdots d_1 d_0$ such that:</p>
<ul><li>$a+bi = d_{n - 1}(i - 1)^{n - 1} + d_{n - 2}(i - 1)^{n - 2} + \cdots + d_1(i - 1) + d_0$</li>
<li>Each $d_k$ is in $\... | 891874596 | Sunday, 22nd March 2015, 10:00 am | 261 | 85% | hard |
19 | Counting Sundays | You are given the following information, but you may prefer to do some research for yourself.
1 Jan 1900 was a Monday.
Thirty days has September,
April, June and November.
All the rest have thirty-one,
Saving February alone,
Which has twenty-eight, rain or shine.
And on leap years, twenty-nine.
A leap year occurs on an... | You are given the following information, but you may prefer to do some research for yourself.
1 Jan 1900 was a Monday.
Thirty days has September,
April, June and November.
All the rest have thirty-one,
Saving February alone,
Which has twenty-eight, rain or shine.
And on leap years, twenty-nine.
A leap year occurs on an... | <p>You are given the following information, but you may prefer to do some research for yourself.</p>
<ul><li>1 Jan 1900 was a Monday.</li>
<li>Thirty days has September,<br>
April, June and November.<br/>
All the rest have thirty-one,<br/>
Saving February alone,<br/>
Which has twenty-eight, rain or shine.<br/>
And on l... | 171 | Friday, 14th June 2002, 06:00 pm | 145831 | 5% | easy |
666 | Polymorphic Bacteria | Members of a species of bacteria occur in two different types: $\alpha$ and $\beta$. Individual bacteria are capable of multiplying and mutating between the types according to the following rules:
Every minute, each individual will simultaneously undergo some kind of transformation.
Each individual $A$ of type $\alpha$... | Members of a species of bacteria occur in two different types: $\alpha$ and $\beta$. Individual bacteria are capable of multiplying and mutating between the types according to the following rules:
Every minute, each individual will simultaneously undergo some kind of transformation.
Each individual $A$ of type $\alpha$... | Members of a species of bacteria occur in two different types: $\alpha$ and $\beta$. Individual bacteria are capable of multiplying and mutating between the types according to the following rules:
<ul><li>Every minute, each individual will simultaneously undergo some kind of transformation.</li>
<li>Each individual $A$... | 0.48023168 | Saturday, 20th April 2019, 01:00 pm | 322 | 45% | medium |
524 | First Sort II | Consider the following algorithm for sorting a list:
1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
2. If the elements are out of order:
a. Move the smallest element of the pair at the beginning of the list.
b. Restart the process from step 1.
3. If all pairs are in order, sto... | Consider the following algorithm for sorting a list:
1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
2. If the elements are out of order:
a. Move the smallest element of the pair at the beginning of the list.
b. Restart the process from step 1.
3. If all pairs are in order, sto... | <p>Consider the following algorithm for sorting a list:</p>
<ul style="list-style-type:none;"><li>1. Starting from the beginning of the list, check each pair of adjacent elements in turn.</li>
<li>2. If the elements are out of order:
<ul style="list-style-type:none;"><li>a. Move the smallest element of the pair at the ... | 2432925835413407847 | Sunday, 6th September 2015, 07:00 am | 247 | 75% | hard |
158 | Lexicographical Neighbours | Taking three different letters from the $26$ letters of the alphabet, character strings of length three can be formed.
Examples are 'abc', 'hat' and 'zyx'.
When we study these three examples we see that for 'abc' two characters come lexicographically after its neighbour to the left.
For 'hat' there is exactly one char... | Taking three different letters from the $26$ letters of the alphabet, character strings of length three can be formed.
Examples are 'abc', 'hat' and 'zyx'.
When we study these three examples we see that for 'abc' two characters come lexicographically after its neighbour to the left.
For 'hat' there is exactly one char... | <p>Taking three different letters from the $26$ letters of the alphabet, character strings of length three can be formed.<br/>
Examples are 'abc', 'hat' and 'zyx'.<br/>
When we study these three examples we see that for 'abc' two characters come lexicographically after its neighbour to the left.<br/>
For 'hat' there i... | 409511334375 | Friday, 15th June 2007, 02:00 pm | 4057 | 55% | medium |
206 | Concealed Square | Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. | Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. | <p>Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0,<br> where each “_” is a single digit.</br></p> | 1389019170 | Saturday, 6th September 2008, 02:00 pm | 26343 | 5% | easy |
336 | Maximix Arrangements | A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order.
To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train moves o... | A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order.
To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train moves o... | <p>A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order. <br>
To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train ... | CAGBIHEFJDK | Saturday, 30th April 2011, 07:00 pm | 2264 | 25% | easy |
459 | Flipping Game | The flipping game is a two player game played on an $N$ by $N$ square board.
Each square contains a disk with one side white and one side black.
The game starts with all disks showing their white side.
A turn consists of flipping all disks in a rectangle with the following properties:
the upper right corner of the rect... | The flipping game is a two player game played on an $N$ by $N$ square board.
Each square contains a disk with one side white and one side black.
The game starts with all disks showing their white side.
A turn consists of flipping all disks in a rectangle with the following properties:
the upper right corner of the rect... | <p>The flipping game is a two player game played on an $N$ by $N$ square board.<br/>
Each square contains a disk with one side white and one side black.<br/>
The game starts with all disks showing their white side.</p>
<p>A turn consists of flipping all disks in a rectangle with the following properties:
</p><p></p><ul... | 3996390106631 | Sunday, 16th February 2014, 10:00 am | 262 | 100% | hard |
224 | Almost Right-angled Triangles II | Let us call an integer sided triangle with sides $a \le b \le c$ barely obtuse if the sides satisfy $a^2 + b^2 = c^2 - 1$.
How many barely obtuse triangles are there with perimeter $\le 75\,000\,000$? | Let us call an integer sided triangle with sides $a \le b \le c$ barely obtuse if the sides satisfy $a^2 + b^2 = c^2 - 1$.
How many barely obtuse triangles are there with perimeter $\le 75\,000\,000$? | <p>Let us call an integer sided triangle with sides $a \le b \le c$ <dfn>barely obtuse</dfn> if the sides satisfy <br/>$a^2 + b^2 = c^2 - 1$.</p>
<p>How many barely obtuse triangles are there with perimeter $\le 75\,000\,000$?</p> | 4137330 | Friday, 26th December 2008, 05:00 pm | 1405 | 75% | hard |
854 | Pisano Periods 2 | For every positive integer $n$ the Fibonacci sequence modulo $n$ is periodic. The period depends on the value of $n$.
This period is called the Pisano period for $n$, often shortened to $\pi(n)$.
Define $M(p)$ as the largest integer $n$ such that $\pi(n) = p$, and define $M(p) = 1$ if there is no such $n$.
For example... | For every positive integer $n$ the Fibonacci sequence modulo $n$ is periodic. The period depends on the value of $n$.
This period is called the Pisano period for $n$, often shortened to $\pi(n)$.
Define $M(p)$ as the largest integer $n$ such that $\pi(n) = p$, and define $M(p) = 1$ if there is no such $n$.
For example... | <p>
For every positive integer $n$ the Fibonacci sequence modulo $n$ is periodic. The period depends on the value of $n$.
This period is called the <strong>Pisano period</strong> for $n$, often shortened to $\pi(n)$.</p>
<p>
Define $M(p)$ as the largest integer $n$ such that $\pi(n) = p$, and define $M(p) = 1$ if there... | 29894398 | Saturday, 9th September 2023, 05:00 pm | 365 | 40% | medium |
579 | Lattice Points in Lattice Cubes | A lattice cube is a cube in which all vertices have integer coordinates. Let $C(n)$ be the number of different lattice cubes in which the coordinates of all vertices range between (and including) $0$ and $n$. Two cubes are hereby considered different if any of their vertices have different coordinates.
For example, $C(... | A lattice cube is a cube in which all vertices have integer coordinates. Let $C(n)$ be the number of different lattice cubes in which the coordinates of all vertices range between (and including) $0$ and $n$. Two cubes are hereby considered different if any of their vertices have different coordinates.
For example, $C(... | <p>A <strong>lattice cube</strong> is a cube in which all vertices have integer coordinates. Let $C(n)$ be the number of different lattice cubes in which the coordinates of all vertices range between (and including) $0$ and $n$. Two cubes are hereby considered different if any of their vertices have different coordinat... | 3805524 | Sunday, 27th November 2016, 01:00 am | 189 | 100% | hard |
522 | Hilbert's Blackout | Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead.
To cut costs, Hilbert wished to power the new hotel with his own special generator. Each floor would send power to the floor above it, with the top floor sending power back down to the bottom floo... | Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead.
To cut costs, Hilbert wished to power the new hotel with his own special generator. Each floor would send power to the floor above it, with the top floor sending power back down to the bottom floo... | <p>Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead.</p>
<p>To cut costs, Hilbert wished to power the new hotel with his own special generator. Each floor would send power to the floor above it, with the top floor sending power back down to the b... | 96772715 | Sunday, 28th June 2015, 04:00 am | 295 | 85% | hard |
363 | Bézier Curves | A cubic Bézier curve is defined by four points: $P_0, P_1, P_2,$ and $P_3$.
The curve is constructed as follows:
On the segments $P_0 P_1$, $P_1 P_2$, and $P_2 P_3$ the points $Q_0, Q_1,$ and $Q_2$ are drawn such that $\dfrac{P_0 Q_0}{P_0 P_1} = \dfrac{P_1 Q_1}{P_1 P_2} = \dfrac{P_2 Q_2}{P_2 P_3} = t$, with $t$ in $[0... | A cubic Bézier curve is defined by four points: $P_0, P_1, P_2,$ and $P_3$.
The curve is constructed as follows:
On the segments $P_0 P_1$, $P_1 P_2$, and $P_2 P_3$ the points $Q_0, Q_1,$ and $Q_2$ are drawn such that $\dfrac{P_0 Q_0}{P_0 P_1} = \dfrac{P_1 Q_1}{P_1 P_2} = \dfrac{P_2 Q_2}{P_2 P_3} = t$, with $t$ in $[0... | <p>A cubic Bézier curve is defined by four points: $P_0, P_1, P_2,$ and $P_3$.</p>
<div class="float_right"><img alt="p363_bezier.png" class="dark_img" src="project/images/p363_bezier.png"/></div>
<p>The curve is constructed as follows:</p>
<p>On the segments $P_0 P_1$, $P_1 P_2$, and $P_2 P_3$ the points $Q_0, Q_1,$ a... | 0.0000372091 | Sunday, 18th December 2011, 10:00 am | 1248 | 35% | medium |
820 | $N$th Digit of Reciprocals | Let $d_n(x)$ be the $n$th decimal digit of the fractional part of $x$, or $0$ if the fractional part has fewer than $n$ digits.
For example:
$d_7 \mathopen{}\left( 1 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 2 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 4 \right)\mathclose{} = d_7 \mathopen{}\left( \... | Let $d_n(x)$ be the $n$th decimal digit of the fractional part of $x$, or $0$ if the fractional part has fewer than $n$ digits.
For example:
$d_7 \mathopen{}\left( 1 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 2 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 4 \right)\mathclose{} = d_7 \mathopen{}\left( \... | <p>Let $d_n(x)$ be the $n$<sup>th</sup> decimal digit of the fractional part of $x$, or $0$ if the fractional part has fewer than $n$ digits.</p>
<p>For example:</p>
<ul>
<li>$d_7 \mathopen{}\left( 1 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 2 \right)\mathclose{} = d_7 \mathopen{}\left( \frac 1 4 \right)\math... | 44967734 | Saturday, 10th December 2022, 04:00 pm | 1060 | 10% | easy |
511 | Sequences with Nice Divisibility Properties | Let $Seq(n,k)$ be the number of positive-integer sequences $\{a_i\}_{1 \le i \le n}$ of length $n$ such that:
$n$ is divisible by $a_i$ for $1 \le i \le n$, and
$n + a_1 + a_2 + \cdots + a_n$ is divisible by $k$.
Examples:
$Seq(3,4) = 4$, and the $4$ sequences are:
$\{1, 1, 3\}$
$\{1, 3, 1\}$
$\{3, 1, 1\}$
$\{3, 3, 3\}... | Let $Seq(n,k)$ be the number of positive-integer sequences $\{a_i\}_{1 \le i \le n}$ of length $n$ such that:
$n$ is divisible by $a_i$ for $1 \le i \le n$, and
$n + a_1 + a_2 + \cdots + a_n$ is divisible by $k$.
Examples:
$Seq(3,4) = 4$, and the $4$ sequences are:
$\{1, 1, 3\}$
$\{1, 3, 1\}$
$\{3, 1, 1\}$
$\{3, 3, 3\}... | <p>Let $Seq(n,k)$ be the number of positive-integer sequences $\{a_i\}_{1 \le i \le n}$ of length $n$ such that:</p>
<ul style="list-style-type:disc;"><li>$n$ is divisible by $a_i$ for $1 \le i \le n$, and</li>
<li>$n + a_1 + a_2 + \cdots + a_n$ is divisible by $k$.</li>
</ul><p>Examples:</p>
<p>$Seq(3,4) = 4$, and the... | 935247012 | Saturday, 11th April 2015, 07:00 pm | 464 | 55% | medium |
572 | Idempotent Matrices | A matrix $M$ is called idempotent if $M^2 = M$.
Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}$.
Let $C(n)$ be the number of idempotent three by three matrices $M$ with integer elements such that
$ -n \le a,b,c,d,e,f,g,h,i \le n$.
$C(1)=164$ and $C(2)=8... | A matrix $M$ is called idempotent if $M^2 = M$.
Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}$.
Let $C(n)$ be the number of idempotent three by three matrices $M$ with integer elements such that
$ -n \le a,b,c,d,e,f,g,h,i \le n$.
$C(1)=164$ and $C(2)=8... | <p>
A matrix $M$ is called idempotent if $M^2 = M$.<br/>
Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}$.<br/>
Let $C(n)$ be the number of idempotent three by three matrices $M$ with integer elements such that<br/>
$ -n \le a,b,c,... | 19737656 | Sunday, 2nd October 2016, 04:00 am | 380 | 50% | medium |
408 | Admissible Paths Through a Grid | Let's call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares.
For example, $(9, 16)$ is inadmissible, while $(0, 4)$, $(3, 1)$ and $(9, 4)$ are not.
Consider a path from point $(x_1, y_1)$ to point $(x_2, y_2)$ using only unit steps north or east.
Let's call such a path admissib... | Let's call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares.
For example, $(9, 16)$ is inadmissible, while $(0, 4)$, $(3, 1)$ and $(9, 4)$ are not.
Consider a path from point $(x_1, y_1)$ to point $(x_2, y_2)$ using only unit steps north or east.
Let's call such a path admissib... | <p>Let's call a lattice point $(x, y)$ <dfn>inadmissible</dfn> if $x, y$ and $x+y$ are all positive perfect squares.<br/>
For example, $(9, 16)$ is inadmissible, while $(0, 4)$, $(3, 1)$ and $(9, 4)$ are not.</p>
<p>Consider a path from point $(x_1, y_1)$ to point $(x_2, y_2)$ using only unit steps north or east.<br/>
... | 299742733 | Saturday, 29th December 2012, 01:00 pm | 640 | 50% | medium |
749 | Near Power Sums | A positive integer, $n$, is a near power sum if there exists a positive integer, $k$, such that the sum of the $k$th powers of the digits in its decimal representation is equal to either $n+1$ or $n-1$. For example $35$ is a near power sum number because $3^2+5^2 = 34$.
Define $S(d)$ to be the sum of all near power s... | A positive integer, $n$, is a near power sum if there exists a positive integer, $k$, such that the sum of the $k$th powers of the digits in its decimal representation is equal to either $n+1$ or $n-1$. For example $35$ is a near power sum number because $3^2+5^2 = 34$.
Define $S(d)$ to be the sum of all near power s... | <p>
A positive integer, $n$, is a <dfn>near power sum</dfn> if there exists a positive integer, $k$, such that the sum of the $k$th powers of the digits in its decimal representation is equal to either $n+1$ or $n-1$. For example $35$ is a near power sum number because $3^2+5^2 = 34$.
</p>
<p>
Define $S(d)$ to be the s... | 13459471903176422 | Saturday, 27th February 2021, 04:00 pm | 798 | 15% | easy |
5 | Smallest Multiple | $2520$ is the smallest number that can be divided by each of the numbers from $1$ to $10$ without any remainder.
What is the smallest positive number that is evenly divisibledivisible with no remainder by all of the numbers from $1$ to $20$? | $2520$ is the smallest number that can be divided by each of the numbers from $1$ to $10$ without any remainder.
What is the smallest positive number that is evenly divisibledivisible with no remainder by all of the numbers from $1$ to $20$? | <p>$2520$ is the smallest number that can be divided by each of the numbers from $1$ to $10$ without any remainder.</p>
<p>What is the smallest positive number that is <strong class="tooltip">evenly divisible<span class="tooltiptext">divisible with no remainder</span></strong> by all of the numbers from $1$ to $20$?</p... | 232792560 | Friday, 30th November 2001, 06:00 pm | 522306 | 5% | easy |
95 | Amicable Chains | The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.
Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of the prope... | The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.
Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of the prope... | <p>The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.</p>
<p>Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of... | 14316 | Friday, 13th May 2005, 06:00 pm | 16208 | 30% | easy |
561 | Divisor Pairs | Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$.
For $n=6$ we get the following pairs: $(1,2), (1,3), (1,6),( 2,6)$ and $(3,6)$. So $S(6)=5$.
Let $p_m\#$ be the product of the first $m$ prime numbers, so $p_2\# = 2*3 = 6$.
Let $E(m, n)$ be the highest integer $k$ such th... | Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$.
For $n=6$ we get the following pairs: $(1,2), (1,3), (1,6),( 2,6)$ and $(3,6)$. So $S(6)=5$.
Let $p_m\#$ be the product of the first $m$ prime numbers, so $p_2\# = 2*3 = 6$.
Let $E(m, n)$ be the highest integer $k$ such th... | <p>
Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$.<br>
For $n=6$ we get the following pairs: $(1,2), (1,3), (1,6),( 2,6)$ and $(3,6)$. So $S(6)=5$.<br/>
Let $p_m\#$ be the product of the first $m$ prime numbers, so $p_2\# = 2*3 = 6$.<br/>
Let $E(m, n)$ be the highest i... | 452480999988235494 | Saturday, 21st May 2016, 10:00 pm | 866 | 30% | easy |
482 | The Incenter of a Triangle | $ABC$ is an integer sided triangle with incenter $I$ and perimeter $p$.
The segments $IA$, $IB$ and $IC$ have integral length as well.
Let $L = p + |IA| + |IB| + |IC|$.
Let $S(P) = \sum L$ for all such triangles where $p \le P$. For example, $S(10^3) = 3619$.
Find $S(10^7)$. | $ABC$ is an integer sided triangle with incenter $I$ and perimeter $p$.
The segments $IA$, $IB$ and $IC$ have integral length as well.
Let $L = p + |IA| + |IB| + |IC|$.
Let $S(P) = \sum L$ for all such triangles where $p \le P$. For example, $S(10^3) = 3619$.
Find $S(10^7)$. | <p>
$ABC$ is an integer sided triangle with incenter $I$ and perimeter $p$.<br/>
The segments $IA$, $IB$ and $IC$ have integral length as well.
</p>
<p>
Let $L = p + |IA| + |IB| + |IC|$.
</p>
<p>
Let $S(P) = \sum L$ for all such triangles where $p \le P$. For example, $S(10^3) = 3619$.
</p>
<p>
Find $S(10^7)$.
</p> | 1400824879147 | Sunday, 28th September 2014, 07:00 am | 247 | 85% | hard |
500 | Problem 500!!! | The number of divisors of $120$ is $16$.
In fact $120$ is the smallest number having $16$ divisors.
Find the smallest number with $2^{500500}$ divisors.
Give your answer modulo $500500507$. | The number of divisors of $120$ is $16$.
In fact $120$ is the smallest number having $16$ divisors.
Find the smallest number with $2^{500500}$ divisors.
Give your answer modulo $500500507$. | <p>The number of divisors of $120$ is $16$.<br/>
In fact $120$ is the smallest number having $16$ divisors.
</p>
<p>
Find the smallest number with $2^{500500}$ divisors.<br/>
Give your answer modulo $500500507$.
</p> | 35407281 | Saturday, 31st January 2015, 01:00 pm | 4493 | 15% | easy |
324 | Building a Tower | Let $f(n)$ represent the number of ways one can fill a $3 \times 3 \times n$ tower with blocks of $2 \times 1 \times 1$.You're allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.
For example (with $q = 100000007$):$f(2) = 229$,$f(4) = 117805... | Let $f(n)$ represent the number of ways one can fill a $3 \times 3 \times n$ tower with blocks of $2 \times 1 \times 1$.You're allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.
For example (with $q = 100000007$):$f(2) = 229$,$f(4) = 117805... | <p>Let $f(n)$ represent the number of ways one can fill a $3 \times 3 \times n$ tower with blocks of $2 \times 1 \times 1$.<br/>You're allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.</p>
<p>For example (with $q = 100000007$):<br/>$f(2) =... | 96972774 | Sunday, 13th February 2011, 10:00 am | 792 | 50% | medium |
94 | Almost Equilateral Triangles | It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle $5$-$5$-$6$ has an area of $12$ square units.
We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than... | It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle $5$-$5$-$6$ has an area of $12$ square units.
We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than... | <p>It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the <dfn>almost equilateral triangle</dfn> $5$-$5$-$6$ has an area of $12$ square units.</p>
<p>We shall define an <dfn>almost equilateral triangle</dfn> to be a triangle for which two sides are equal and t... | 518408346 | Friday, 29th April 2005, 06:00 pm | 14113 | 35% | medium |
771 | Pseudo Geometric Sequences | We define a pseudo-geometric sequence to be a finite sequence $a_0, a_1, \dotsc, a_n$ of positive integers, satisfying the following conditions:
$n \geq 4$, i.e. the sequence has at least $5$ terms.
$0 \lt a_0 \lt a_1 \lt \cdots \lt a_n$, i.e. the sequence is strictly increasing.
$| a_i^2 - a_{i - 1}a_{i + 1} | \le 2$... | We define a pseudo-geometric sequence to be a finite sequence $a_0, a_1, \dotsc, a_n$ of positive integers, satisfying the following conditions:
$n \geq 4$, i.e. the sequence has at least $5$ terms.
$0 \lt a_0 \lt a_1 \lt \cdots \lt a_n$, i.e. the sequence is strictly increasing.
$| a_i^2 - a_{i - 1}a_{i + 1} | \le 2$... | <p>
We define a <dfn>pseudo-geometric sequence</dfn> to be a finite sequence $a_0, a_1, \dotsc, a_n$ of positive integers, satisfying the following conditions:
</p><ul>
<li>$n \geq 4$, i.e. the sequence has at least $5$ terms.</li>
<li>$0 \lt a_0 \lt a_1 \lt \cdots \lt a_n$, i.e. the sequence is strictly increasing.</l... | 398803409 | Saturday, 6th November 2021, 01:00 pm | 135 | 80% | hard |
779 | Prime Factor and Exponent | For a positive integer $n \gt 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha(n)$ be its $p$-adic order, i.e. the largest integer such that $p(n)^{\alpha(n)}$ divides $n$.
For a positive integer $K$, define the function $f_K(n)$ by:
$$f_K(n)=\frac{\alpha(n)-1}{(p(n))^K}.$$
Also define $\overline{f... | For a positive integer $n \gt 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha(n)$ be its $p$-adic order, i.e. the largest integer such that $p(n)^{\alpha(n)}$ divides $n$.
For a positive integer $K$, define the function $f_K(n)$ by:
$$f_K(n)=\frac{\alpha(n)-1}{(p(n))^K}.$$
Also define $\overline{f... | <p>
For a positive integer $n \gt 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha(n)$ be its <strong>$p$-adic order</strong>, i.e. the largest integer such that $p(n)^{\alpha(n)}$ divides $n$.
</p>
<p>
For a positive integer $K$, define the function $f_K(n)$ by:
$$f_K(n)=\frac{\alpha(n)-1}{(p(n))^K}.... | 0.547326103833 | Saturday, 1st January 2022, 01:00 pm | 535 | 25% | easy |
390 | Triangles with Non Rational Sides and Integral Area | Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$.
It can be shown that this triangle has area $9$.
$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (for positive integers $b$ and $c$) that have an integral area not exceeding $n$.... | Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$.
It can be shown that this triangle has area $9$.
$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (for positive integers $b$ and $c$) that have an integral area not exceeding $n$.... | <p>Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$.
It can be shown that this triangle has area $9$.</p>
<p>$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (for positive integers $b$ and $c$) that have an integral area not exce... | 2919133642971 | Saturday, 23rd June 2012, 08:00 pm | 625 | 60% | hard |
398 | Cutting Rope | Inside a rope of length $n$, $n - 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose $m - 1$ points at random and cut the rope at these points to create $m$ segments.
Let $E(n, m)$ be the expected length of the second-shortest segment.
For example, $E(3, 2) = ... | Inside a rope of length $n$, $n - 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose $m - 1$ points at random and cut the rope at these points to create $m$ segments.
Let $E(n, m)$ be the expected length of the second-shortest segment.
For example, $E(3, 2) = ... | <p>
Inside a rope of length $n$, $n - 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose $m - 1$ points at random and cut the rope at these points to create $m$ segments.
</p>
<p>
Let $E(n, m)$ be the expected length of the second-shortest segment.
For example, ... | 2010.59096 | Sunday, 14th October 2012, 08:00 am | 425 | 55% | medium |
565 | Divisibility of Sum of Divisors | Let $\sigma(n)$ be the sum of the divisors of $n$.
E.g. the divisors of $4$ are $1$, $2$ and $4$, so $\sigma(4)=7$.
The numbers $n$ not exceeding $20$ such that $7$ divides $\sigma(n)$ are: $4$, $12$, $13$ and $20$, the sum of these numbers being $49$.
Let $S(n, d)$ be the sum of the numbers $i$ not exceeding $n$... | Let $\sigma(n)$ be the sum of the divisors of $n$.
E.g. the divisors of $4$ are $1$, $2$ and $4$, so $\sigma(4)=7$.
The numbers $n$ not exceeding $20$ such that $7$ divides $\sigma(n)$ are: $4$, $12$, $13$ and $20$, the sum of these numbers being $49$.
Let $S(n, d)$ be the sum of the numbers $i$ not exceeding $n$... | <p>Let $\sigma(n)$ be the sum of the divisors of $n$.<br/>
E.g. the divisors of $4$ are $1$, $2$ and $4$, so $\sigma(4)=7$.
</p>
<p>
The numbers $n$ not exceeding $20$ such that $7$ divides $\sigma(n)$ are: $4$, $12$, $13$ and $20$, the sum of these numbers being $49$.
</p>
<p>
Let $S(n, d)$ be the sum of the numbers... | 2992480851924313898 | Sunday, 19th June 2016, 10:00 am | 700 | 35% | medium |
33 | Digit Cancelling Fractions | The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.
We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.
There are exactly four non-trivial example... | The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.
We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.
There are exactly four non-trivial example... | <p>The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.</p>
<p>We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.</p>
<p>There are exactly four no... | 100 | Friday, 20th December 2002, 06:00 pm | 78466 | 5% | easy |
387 | Harshad Numbers | A Harshad or Niven number is a number that is divisible by the sum of its digits.
$201$ is a Harshad number because it is divisible by $3$ (the sum of its digits.)
When we truncate the last digit from $201$, we get $20$, which is a Harshad number.
When we truncate the last digit from $20$, we get $2$, which is also a H... | A Harshad or Niven number is a number that is divisible by the sum of its digits.
$201$ is a Harshad number because it is divisible by $3$ (the sum of its digits.)
When we truncate the last digit from $201$, we get $20$, which is a Harshad number.
When we truncate the last digit from $20$, we get $2$, which is also a H... | <p>A <strong>Harshad or Niven number</strong> is a number that is divisible by the sum of its digits.
<br/>$201$ is a Harshad number because it is divisible by $3$ (the sum of its digits.)
<br/>When we truncate the last digit from $201$, we get $20$, which is a Harshad number.
<br/>When we truncate the last digit from ... | 696067597313468 | Sunday, 3rd June 2012, 11:00 am | 5096 | 10% | easy |
365 | A Huge Binomial Coefficient | The binomial coefficient $\displaystyle{\binom{10^{18}}{10^9}}$ is a number with more than $9$ billion ($9\times 10^9$) digits.
Let $M(n,k,m)$ denote the binomial coefficient $\displaystyle{\binom{n}{k}}$ modulo $m$.
Calculate $\displaystyle{\sum M(10^{18},10^9,p\cdot q\cdot r)}$ for $1000\lt p\lt q\lt r\lt 5000$ a... | The binomial coefficient $\displaystyle{\binom{10^{18}}{10^9}}$ is a number with more than $9$ billion ($9\times 10^9$) digits.
Let $M(n,k,m)$ denote the binomial coefficient $\displaystyle{\binom{n}{k}}$ modulo $m$.
Calculate $\displaystyle{\sum M(10^{18},10^9,p\cdot q\cdot r)}$ for $1000\lt p\lt q\lt r\lt 5000$ a... | <p>
The binomial coefficient $\displaystyle{\binom{10^{18}}{10^9}}$ is a number with more than $9$ billion ($9\times 10^9$) digits.
</p>
<p>
Let $M(n,k,m)$ denote the binomial coefficient $\displaystyle{\binom{n}{k}}$ modulo $m$.
</p>
<p>
Calculate $\displaystyle{\sum M(10^{18},10^9,p\cdot q\cdot r)}$ for $1000\lt p\lt... | 162619462356610313 | Saturday, 31st December 2011, 04:00 pm | 1436 | 40% | medium |
668 | Square Root Smooth Numbers | A positive integer is called square root smooth if all of its prime factors are strictly less than its square root.
Including the number $1$, there are $29$ square root smooth numbers not exceeding $100$.
How many square root smooth numbers are there not exceeding $10\,000\,000\,000$? | A positive integer is called square root smooth if all of its prime factors are strictly less than its square root.
Including the number $1$, there are $29$ square root smooth numbers not exceeding $100$.
How many square root smooth numbers are there not exceeding $10\,000\,000\,000$? | <p>
A positive integer is called <dfn>square root smooth</dfn> if all of its prime factors are strictly less than its square root.<br/>
Including the number $1$, there are $29$ square root smooth numbers not exceeding $100$.
</p>
<p>
How many square root smooth numbers are there not exceeding $10\,000\,000\,000$?</p> | 2811077773 | Saturday, 4th May 2019, 07:00 pm | 1173 | 20% | easy |
454 | Diophantine Reciprocals III | In the following equation $x$, $y$, and $n$ are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$
For a limit $L$ we define $F(L)$ as the number of solutions which satisfy $x \lt y \le L$.
We can verify that $F(15) = 4$ and $F(1000) = 1069$.
Find $F(10^{12})$. | In the following equation $x$, $y$, and $n$ are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$
For a limit $L$ we define $F(L)$ as the number of solutions which satisfy $x \lt y \le L$.
We can verify that $F(15) = 4$ and $F(1000) = 1069$.
Find $F(10^{12})$. | <p>In the following equation $x$, $y$, and $n$ are positive integers.</p>
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$
<p>For a limit $L$ we define $F(L)$ as the number of solutions which satisfy $x \lt y \le L$.</p>
<p>We can verify that $F(15) = 4$ and $F(1000) = 1069$.<br/>
Find $F(10^{12})$.</p> | 5435004633092 | Saturday, 11th January 2014, 07:00 pm | 534 | 50% | medium |
441 | The Inverse Summation of Coprime Couples | For an integer $M$, we define $R(M)$ as the sum of $1/(p \cdot q)$ for all the integer pairs $p$ and $q$ which satisfy all of these conditions:
$1 \leq p \lt q \leq M$
$p + q \geq M$
$p$ and $q$ are coprime.
We also define $S(N)$ as the sum of $R(i)$ for $2 \leq i \leq N$.
We can verify that $S(2) = R(2) = 1/2$, $... | For an integer $M$, we define $R(M)$ as the sum of $1/(p \cdot q)$ for all the integer pairs $p$ and $q$ which satisfy all of these conditions:
$1 \leq p \lt q \leq M$
$p + q \geq M$
$p$ and $q$ are coprime.
We also define $S(N)$ as the sum of $R(i)$ for $2 \leq i \leq N$.
We can verify that $S(2) = R(2) = 1/2$, $... | <p>
For an integer $M$, we define $R(M)$ as the sum of $1/(p \cdot q)$ for all the integer pairs $p$ and $q$ which satisfy all of these conditions:
</p>
<ul><li> $1 \leq p \lt q \leq M$</li>
<li> $p + q \geq M$</li>
<li> $p$ and $q$ are coprime.</li>
</ul><p>
We also define $S(N)$ as the sum of $R(i)$ for $2 \leq i \le... | 5000088.8395 | Sunday, 20th October 2013, 10:00 am | 396 | 65% | hard |
641 | A Long Row of Dice | Consider a row of $n$ dice all showing 1.
First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every third die. The sixth die will now show a 3. Then turn every fourth die and so on until every $n$th die (only the last die) is turned. If the die to be turned is showing ... | Consider a row of $n$ dice all showing 1.
First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every third die. The sixth die will now show a 3. Then turn every fourth die and so on until every $n$th die (only the last die) is turned. If the die to be turned is showing ... | <p>Consider a row of $n$ dice all showing 1.</p>
<p>First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every third die. The sixth die will now show a 3. Then turn every fourth die and so on until every $n$th die (only the last die) is turned. If the die to be turned i... | 793525366 | Saturday, 3rd November 2018, 01:00 pm | 549 | 35% | medium |
634 | Numbers of the Form $a^2b^3$ | Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and both greater than 1.
For example, $32=2^2\times 2^3$ and $72=3^2\times 2^3$ are the only two integers less than $100$ that can be written in this form. Hence, $F(10... | Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and both greater than 1.
For example, $32=2^2\times 2^3$ and $72=3^2\times 2^3$ are the only two integers less than $100$ that can be written in this form. Hence, $F(10... | <p>
Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and both greater than 1.</p>
<p>
For example, $32=2^2\times 2^3$ and $72=3^2\times 2^3$ are the only two integers less than $100$ that can be written in this form. H... | 4019680944 | Saturday, 11th August 2018, 04:00 pm | 690 | 40% | medium |
228 | Minkowski Sums | Let $S_n$ be the regular $n$-sided polygon – or shape – whose vertices
$v_k$ ($k = 1, 2, \dots, n$) have coordinates:
\begin{align}
x_k &= \cos((2k - 1)/n \times 180^\circ)\\
y_k &= \sin((2k - 1)/n \times 180^\circ)
\end{align}
Each $S_n$ is to be interpreted as a filled shape consisting of all points on the perimet... | Let $S_n$ be the regular $n$-sided polygon – or shape – whose vertices
$v_k$ ($k = 1, 2, \dots, n$) have coordinates:
\begin{align}
x_k &= \cos((2k - 1)/n \times 180^\circ)\\
y_k &= \sin((2k - 1)/n \times 180^\circ)
\end{align}
Each $S_n$ is to be interpreted as a filled shape consisting of all points on the perimet... | <p>Let $S_n$ be the regular $n$-sided polygon – or <dfn>shape</dfn> – whose vertices
$v_k$ ($k = 1, 2, \dots, n$) have coordinates:</p>
\begin{align}
x_k &= \cos((2k - 1)/n \times 180^\circ)\\
y_k &= \sin((2k - 1)/n \times 180^\circ)
\end{align}
<p>Each $S_n$ is to be interpreted as a filled shape consisting of all ... | 86226 | Saturday, 17th January 2009, 05:00 am | 1494 | 70% | hard |
370 | Geometric Triangles | Let us define a geometric triangle as an integer sided triangle with sides $a \le b \le c$ so that its sides form a geometric progression, i.e. $b^2 = a \cdot c$
An example of such a geometric triangle is the triangle with sides $a = 144$, $b = 156$ and $c = 169$.
There are $861805$ geometric triangles with perimeter $... | Let us define a geometric triangle as an integer sided triangle with sides $a \le b \le c$ so that its sides form a geometric progression, i.e. $b^2 = a \cdot c$
An example of such a geometric triangle is the triangle with sides $a = 144$, $b = 156$ and $c = 169$.
There are $861805$ geometric triangles with perimeter $... | <p>Let us define a <dfn>geometric triangle</dfn> as an integer sided triangle with sides $a \le b \le c$ so that its sides form a <strong>geometric progression</strong>, i.e. $b^2 = a \cdot c$</p>
<p>An example of such a geometric triangle is the triangle with sides $a = 144$, $b = 156$ and $c = 169$.</p>
<p>There are ... | 41791929448408 | Sunday, 5th February 2012, 07:00 am | 592 | 65% | hard |
643 | $2$-Friendly | Two positive integers $a$ and $b$ are $2$-friendly when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 = 2^3$ while $24$ and $36$ are not because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of $2$.
Let $f(n)$ be the number of pairs, $(p,q)$, of positive integers with $1... | Two positive integers $a$ and $b$ are $2$-friendly when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 = 2^3$ while $24$ and $36$ are not because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of $2$.
Let $f(n)$ be the number of pairs, $(p,q)$, of positive integers with $1... | <p>Two positive integers $a$ and $b$ are <dfn>$2$-friendly</dfn> when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 = 2^3$ while $24$ and $36$ are not because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of $2$.</p>
<p>Let $f(n)$ be the number of pairs, $(p,q)$, of posi... | 968274154 | Saturday, 17th November 2018, 07:00 pm | 694 | 25% | easy |
417 | Reciprocal Cycles II | A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:
\begin{align}
1/2 &= 0.5\\
1/3 &=0.(3)\\
1/4 &=0.25\\
1/5 &= 0.2\\
1/6 &= 0.1(6)\\
1/7 &= 0.(142857)\\
1/8 &= 0.125\\
1/9 &= 0.(1)\\
1/10 &= 0.1
\end{align}
Where $0.1(6)$ means $0.1... | A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:
\begin{align}
1/2 &= 0.5\\
1/3 &=0.(3)\\
1/4 &=0.25\\
1/5 &= 0.2\\
1/6 &= 0.1(6)\\
1/7 &= 0.(142857)\\
1/8 &= 0.125\\
1/9 &= 0.(1)\\
1/10 &= 0.1
\end{align}
Where $0.1(6)$ means $0.1... | <p>A unit fraction contains $1$ in the numerator. The decimal representation of the unit fractions with denominators $2$ to $10$ are given:</p>
\begin{align}
1/2 &= 0.5\\
1/3 &=0.(3)\\
1/4 &=0.25\\
1/5 &= 0.2\\
1/6 &= 0.1(6)\\
1/7 &= 0.(142857)\\
1/8 &= 0.125\\
1/9 &= 0.(1)\\
1/10 &= 0.1
\end{align}
<p>Where $0.1(6)$ ... | 446572970925740 | Saturday, 2nd March 2013, 04:00 pm | 993 | 35% | medium |
453 | Lattice Quadrilaterals | A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.
Let $Q(m, n)$ be the number of simple quadrilaterals whose vertices are lattice points with coordinates $(x,y)$ satisfying $0 \le x \le m$ and $0 \le y \le n$.
For example, $Q(2, 2) = 94$ as can be s... | A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.
Let $Q(m, n)$ be the number of simple quadrilaterals whose vertices are lattice points with coordinates $(x,y)$ satisfying $0 \le x \le m$ and $0 \le y \le n$.
For example, $Q(2, 2) = 94$ as can be s... | <p>A <strong>simple quadrilateral</strong> is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.</p>
<p>Let $Q(m, n)$ be the number of simple quadrilaterals whose vertices are lattice points with coordinates $(x,y)$ satisfying $0 \le x \le m$ and $0 \le y \le n$.</p>
<p>For e... | 104354107 | Saturday, 4th January 2014, 04:00 pm | 239 | 95% | hard |
82 | Path Sum: Three Ways | NOTE: This problem is a more challenging version of Problem 81.
The minimal path sum in the $5$ by $5$ matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red and bold; the sum is equal to $994$.
$$
\begin{pmatrix}... | NOTE: This problem is a more challenging version of Problem 81.
The minimal path sum in the $5$ by $5$ matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red and bold; the sum is equal to $994$.
$$
\begin{pmatrix}... | <p class="small_notice">NOTE: This problem is a more challenging version of <a href="problem=81">Problem 81</a>.</p>
<p>The minimal path sum in the $5$ by $5$ matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red ... | 260324 | Friday, 5th November 2004, 06:00 pm | 23449 | 20% | easy |
755 | Not Zeckendorf | Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$.
We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers.
For example, $16 = 3+13 = 1+2+13 = 3+5+8 = 1+2+5+8$ and hence $f(16) = 4$.
By convention $f(0) = 1$.
Further we define
$$S(n) = \sum_{k=0}... | Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$.
We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers.
For example, $16 = 3+13 = 1+2+13 = 3+5+8 = 1+2+5+8$ and hence $f(16) = 4$.
By convention $f(0) = 1$.
Further we define
$$S(n) = \sum_{k=0}... | <p>
Consider the Fibonacci sequence $\{1,2,3,5,8,13,21,\ldots\}$.
</p>
<p>
We let $f(n)$ be the number of ways of representing an integer $n\ge 0$ as the sum of different Fibonacci numbers.<br/>
For example, $16 = 3+13 = 1+2+13 = 3+5+8 = 1+2+5+8$ and hence $f(16) = 4$.
By convention $f(0) = 1$.
</p>
<p>
Further we def... | 2877071595975576960 | Sunday, 18th April 2021, 11:00 am | 972 | 15% | easy |
790 | Clock Grid | There is a grid of length and width $50515093$ points. A clock is placed on each grid point. The clocks are all analogue showing a single hour hand initially pointing at $12$.
A sequence $S_t$ is created where:
$$
\begin{align}
S_0 &= 290797\\
S_t &= S_{t-1}^2 \bmod 50515093 &t>0
\end{align}
$$
The four numbers $N_t = ... | There is a grid of length and width $50515093$ points. A clock is placed on each grid point. The clocks are all analogue showing a single hour hand initially pointing at $12$.
A sequence $S_t$ is created where:
$$
\begin{align}
S_0 &= 290797\\
S_t &= S_{t-1}^2 \bmod 50515093 &t>0
\end{align}
$$
The four numbers $N_t = ... | <p>There is a grid of length and width $50515093$ points. A clock is placed on each grid point. The clocks are all analogue showing a single hour hand initially pointing at $12$.</p>
<p>A sequence $S_t$ is created where:
$$
\begin{align}
S_0 &= 290797\\
S_t &= S_{t-1}^2 \bmod 50515093 &t>0
\end{align}
$$... | 16585056588495119 | Saturday, 19th March 2022, 10:00 pm | 281 | 45% | medium |
371 | Licence Plates | Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).
While driving to work Seth plays the following game:
Whenever the numbers of two licence plates seen on his trip add to 1000 that's a win.
E.g. MIC-012 and HAN-988 is a win and RYU-500 and SET-500 too (as ... | Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).
While driving to work Seth plays the following game:
Whenever the numbers of two licence plates seen on his trip add to 1000 that's a win.
E.g. MIC-012 and HAN-988 is a win and RYU-500 and SET-500 too (as ... | <p>
Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).<br/>
While driving to work Seth plays the following game:<br/>
Whenever the numbers of two licence plates seen on his trip add to 1000 that's a win.
</p>
<p>
E.g. MIC-012 and HAN-988 is a win and RYU-500... | 40.66368097 | Sunday, 12th February 2012, 10:00 am | 1819 | 30% | easy |
312 | Cyclic Paths on Sierpiński Graphs | - A Sierpiński graph of order-$1$ ($S_1$) is an equilateral triangle.
- $S_{n + 1}$ is obtained from $S_n$ by positioning three copies of $S_n$ so that every pair of copies has one common corner.
Let $C(n)$ be the number of cycles that pass exactly once through all the vertices of $S_n$.
For example, $C(3) = 8$ becau... | - A Sierpiński graph of order-$1$ ($S_1$) is an equilateral triangle.
- $S_{n + 1}$ is obtained from $S_n$ by positioning three copies of $S_n$ so that every pair of copies has one common corner.
Let $C(n)$ be the number of cycles that pass exactly once through all the vertices of $S_n$.
For example, $C(3) = 8$ becau... | <p>- A <strong>Sierpiński graph</strong> of order-$1$ ($S_1$) is an equilateral triangle.<br/>
- $S_{n + 1}$ is obtained from $S_n$ by positioning three copies of $S_n$ so that every pair of copies has one common corner.
</p>
<div align="center"><img alt="0312_sierpinskyAt.gif" class="dark_img" src="resources/images/03... | 324681947 | Sunday, 28th November 2010, 01:00 am | 944 | 50% | medium |
451 | Modular Inverses | Consider the number $15$.
There are eight positive numbers less than $15$ which are coprime to $15$: $1, 2, 4, 7, 8, 11, 13, 14$.
The modular inverses of these numbers modulo $15$ are: $1, 8, 4, 13, 2, 11, 7, 14$
because
$1 \cdot 1 \bmod 15=1$
$2 \cdot 8=16 \bmod 15=1$
$4 \cdot 4=16 \bmod 15=1$
$7 \cdot 13=91 \bmod 1... | Consider the number $15$.
There are eight positive numbers less than $15$ which are coprime to $15$: $1, 2, 4, 7, 8, 11, 13, 14$.
The modular inverses of these numbers modulo $15$ are: $1, 8, 4, 13, 2, 11, 7, 14$
because
$1 \cdot 1 \bmod 15=1$
$2 \cdot 8=16 \bmod 15=1$
$4 \cdot 4=16 \bmod 15=1$
$7 \cdot 13=91 \bmod 1... | <p>
Consider the number $15$.<br/>
There are eight positive numbers less than $15$ which are coprime to $15$: $1, 2, 4, 7, 8, 11, 13, 14$.<br/>
The modular inverses of these numbers modulo $15$ are: $1, 8, 4, 13, 2, 11, 7, 14$<br/>
because<br/>
$1 \cdot 1 \bmod 15=1$<br/>
$2 \cdot 8=16 \bmod 15=1$<br/>
$4 \cdot 4=16 ... | 153651073760956 | Sunday, 22nd December 2013, 10:00 am | 1630 | 30% | easy |
53 | Combinatoric Selections | There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.
In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \times (n-1) \times ... \times 3 \times 2 \... | There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.
In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \times (n-1) \times ... \times 3 \times 2 \... | <p>There are exactly ten ways of selecting three from five, 12345:</p>
<p class="center">123, 124, 125, 134, 135, 145, 234, 235, 245, and 345</p>
<p>In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.</p>
<p>In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \tim... | 4075 | Friday, 26th September 2003, 06:00 pm | 64266 | 5% | easy |
837 | Amidakuji | Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.
In the beginning, a number of parallel vertical lines are drawn, one for each object. Then a specified number of horizontal rungs are added, each lower than any previous rungs. Each rung is drawn as a line segment spanning a... | Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.
In the beginning, a number of parallel vertical lines are drawn, one for each object. Then a specified number of horizontal rungs are added, each lower than any previous rungs. Each rung is drawn as a line segment spanning a... | <p>
<a href="https://en.wikipedia.org/wiki/Amidakuji">Amidakuji</a> (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.</p>
<p>
In the beginning, a number of parallel vertical lines are drawn, one for each object. Then a specified number of horizontal rungs are added, each lower than an... | 428074856 | Saturday, 1st April 2023, 05:00 pm | 221 | 55% | medium |
343 | Fractional Sequences | For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by:
$a_1 = 1/k$ and
$a_i = (x_{i - 1} + 1) / (y_{i - 1} - 1)$ reduced to lowest terms for $i \gt 1$.
When $a_i$ reaches some integer $n$, the sequence stops. (That is, when $y_i = 1$.)
Define $f(k) = n$.
For example, for $k = 20$:
... | For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by:
$a_1 = 1/k$ and
$a_i = (x_{i - 1} + 1) / (y_{i - 1} - 1)$ reduced to lowest terms for $i \gt 1$.
When $a_i$ reaches some integer $n$, the sequence stops. (That is, when $y_i = 1$.)
Define $f(k) = n$.
For example, for $k = 20$:
... | <p>For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by:<br/>
$a_1 = 1/k$ and<br/>
$a_i = (x_{i - 1} + 1) / (y_{i - 1} - 1)$ reduced to lowest terms for $i \gt 1$.<br/>
When $a_i$ reaches some integer $n$, the sequence stops. (That is, when $y_i = 1$.)<br/>
Define $f(k) = n$. <br/>... | 269533451410884183 | Saturday, 18th June 2011, 04:00 pm | 1552 | 35% | medium |
541 | Divisibility of Harmonic Number Denominators | The $n$th harmonic number $H_n$ is defined as the sum of the multiplicative inverses of the first $n$ positive integers, and can be written as a reduced fraction $a_n/b_n$.
$H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}$, with $\gcd(a_n, b_n)=1$.
Let $M(p)$ be the largest value of $n$ such that $b_n$ i... | The $n$th harmonic number $H_n$ is defined as the sum of the multiplicative inverses of the first $n$ positive integers, and can be written as a reduced fraction $a_n/b_n$.
$H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}$, with $\gcd(a_n, b_n)=1$.
Let $M(p)$ be the largest value of $n$ such that $b_n$ i... | <p>The $n$<sup>th</sup> <strong>harmonic number</strong> $H_n$ is defined as the sum of the multiplicative inverses of the first $n$ positive integers, and can be written as a <strong>reduced fraction</strong> $a_n/b_n$.<br/>
$H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}$, with $\gcd(a_n, b_n)=1$.</p>... | 4580726482872451 | Sunday, 3rd January 2016, 10:00 am | 233 | 90% | hard |
268 | At Least Four Distinct Prime Factors Less Than 100 | It can be verified that there are $23$ positive integers less than $1000$ that are divisible by at least four distinct primes less than $100$.
Find how many positive integers less than $10^{16}$ are divisible by at least four distinct primes less than $100$. | It can be verified that there are $23$ positive integers less than $1000$ that are divisible by at least four distinct primes less than $100$.
Find how many positive integers less than $10^{16}$ are divisible by at least four distinct primes less than $100$. | <p>It can be verified that there are $23$ positive integers less than $1000$ that are divisible by at least four distinct primes less than $100$.</p>
<p>Find how many positive integers less than $10^{16}$ are divisible by at least four distinct primes less than $100$.</p> | 785478606870985 | Friday, 11th December 2009, 05:00 pm | 1627 | 70% | hard |
722 | Slowly Converging Series | For a non-negative integer $k$, define
\[
E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n
\]
where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th powers of the positive divisors of $n$.
It can be shown that, for every $k$, the series $E_k(q)$ converges for any $0 < q < 1$.
For example,
$E_1(1 - \frac{1... | For a non-negative integer $k$, define
\[
E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n
\]
where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th powers of the positive divisors of $n$.
It can be shown that, for every $k$, the series $E_k(q)$ converges for any $0 < q < 1$.
For example,
$E_1(1 - \frac{1... | <p>For a non-negative integer $k$, define
\[
E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n
\]
where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th powers of the positive divisors of $n$.</p>
<p>It can be shown that, for every $k$, the series $E_k(q)$ converges for any $0 < q < 1$.</p>
<p>For ex... | 3.376792776502e132 | Sunday, 28th June 2020, 05:00 am | 571 | 25% | easy |
107 | Minimal Network | The following undirected network consists of seven vertices and twelve edges with a total weight of 243.
The same network can be represented by the matrix below.
ABCDEFG
A-161221---
B16--1720--
C12--28-31-
D211728-181923
E-20-18--11
F--3119--27
G---231127-
However, it is possible to optimise the network by removi... | The following undirected network consists of seven vertices and twelve edges with a total weight of 243.
The same network can be represented by the matrix below.
ABCDEFG
A-161221---
B16--1720--
C12--28-31-
D211728-181923
E-20-18--11
F--3119--27
G---231127-
However, it is possible to optimise the network by removi... | <p>The following undirected network consists of seven vertices and twelve edges with a total weight of 243.</p>
<div class="center">
<img alt="" class="dark_img" src="resources/images/0107_1.png?1678992052"/><br/></div>
<p>The same network can be represented by the matrix below.</p>
<table align="center" border="1" cel... | 259679 | Friday, 21st October 2005, 06:00 pm | 12252 | 35% | medium |
123 | Prime Square Remainders | Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n - 1)^n + (p_n + 1)^n$ is divided by $p_n^2$.
For example, when $n = 3$, $p_3 = 5$, and $4^3 + 6^3 = 280 \equiv 5 \mod 25$.
The least value of $n$ for which the remainder first exceeds $10^9$ is $7037$.
Find the least value of... | Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n - 1)^n + (p_n + 1)^n$ is divided by $p_n^2$.
For example, when $n = 3$, $p_3 = 5$, and $4^3 + 6^3 = 280 \equiv 5 \mod 25$.
The least value of $n$ for which the remainder first exceeds $10^9$ is $7037$.
Find the least value of... | <p>Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n - 1)^n + (p_n + 1)^n$ is divided by $p_n^2$.</p>
<p>For example, when $n = 3$, $p_3 = 5$, and $4^3 + 6^3 = 280 \equiv 5 \mod 25$.</p>
<p>The least value of $n$ for which the remainder first exceeds $10^9$ is $7037$.</p>
<p... | 21035 | Friday, 16th June 2006, 06:00 pm | 12733 | 30% | easy |
397 | Triangle on Parabola | On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen.
Let $F(K, X)$ be the number of the integer quadruplets $(k, a, b, c)$ such that at least one angle of the triangle $ABC$ is $45$-degree, with $1 \le k \le K$ and $-X \le a \lt b \lt c \le X$.
For example, $F(1, 10) ... | On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen.
Let $F(K, X)$ be the number of the integer quadruplets $(k, a, b, c)$ such that at least one angle of the triangle $ABC$ is $45$-degree, with $1 \le k \le K$ and $-X \le a \lt b \lt c \le X$.
For example, $F(1, 10) ... | <p>
On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen.
</p>
<p>
Let $F(K, X)$ be the number of the integer quadruplets $(k, a, b, c)$ such that at least one angle of the triangle $ABC$ is $45$-degree, with $1 \le k \le K$ and $-X \le a \lt b \lt c \le X$.
</p>
<p>
For e... | 141630459461893728 | Sunday, 7th October 2012, 05:00 am | 301 | 70% | hard |
922 | Young's Game A | A Young diagram is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such that
the left-most squares of all rows are aligned vertically;
the top squares of all columns are aligned horizontally;
the rows are non-increasing in size as we move top to bottom;
the columns are no... | A Young diagram is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such that
the left-most squares of all rows are aligned vertically;
the top squares of all columns are aligned horizontally;
the rows are non-increasing in size as we move top to bottom;
the columns are no... | <p>
A <strong>Young diagram</strong> is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such that</p>
<ul>
<li>the left-most squares of all rows are aligned vertically;
</li><li>the top squares of all columns are aligned horizontally;
</li><li>the rows are non-increasing i... | 858945298 | Saturday, 21st December 2024, 04:00 pm | 118 | 70% | hard |
534 | Weak Queens | The classical eight queens puzzle is the well known problem of placing eight chess queens on an $8 \times 8$ chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of $92$ distinct configurations can be found for eight queens. The general case asks... | The classical eight queens puzzle is the well known problem of placing eight chess queens on an $8 \times 8$ chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of $92$ distinct configurations can be found for eight queens. The general case asks... | <p>The classical <b>eight queens puzzle</b> is the well known problem of placing eight chess queens on an $8 \times 8$ chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of $92$ distinct configurations can be found for eight queens. The general... | 11726115562784664 | Saturday, 14th November 2015, 01:00 pm | 344 | 55% | medium |
731 | A Stoneham Number | $$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$
Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward.
For example, $A(100) = 4938271604$ and $A(10^8)=2584642393$.
Find $A(10^{16})$. | $$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$
Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward.
For example, $A(100) = 4938271604$ and $A(10^8)=2584642393$.
Find $A(10^{16})$. | <p>
$$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$
</p>
<p>
Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward.
For example, $A(100) = 4938271604$ and $A(10^8)=2584642393$.
</p>
<p>
Find $A(10^{16})$.
</p> | 6086371427 | Sunday, 25th October 2020, 10:00 am | 637 | 20% | easy |
385 | Ellipses Inside Triangles | For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.
For a given $n$, consider triangles $T$ such that:
- the vertices of $T$ have integer coordinates with absolute value $\le n$, and
- the foci1 of the largest-area ellipse inside $T$ are... | For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.
For a given $n$, consider triangles $T$ such that:
- the vertices of $T$ have integer coordinates with absolute value $\le n$, and
- the foci1 of the largest-area ellipse inside $T$ are... | <p>
For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.
</p><p align="center">
<img alt="0385_ellipsetriangle.png" src="resources/images/0385_ellipsetriangle.png?1678992053"/></p>
<p>
For a given $n$, consider triangles $T$ such that:<br/>
-... | 3776957309612153700 | Sunday, 20th May 2012, 05:00 am | 318 | 70% | hard |
106 | Special Subset Sums: Meta-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 this ... | 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 this ... | <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... | 21384 | Friday, 7th October 2005, 06:00 pm | 7291 | 50% | medium |
267 | Billionaire | You are given a unique investment opportunity.
Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.
Your return is double your bet for heads and you lose your bet for tails.
For example, if f = 1/4, for the first toss you bet £0.25, a... | You are given a unique investment opportunity.
Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.
Your return is double your bet for heads and you lose your bet for tails.
For example, if f = 1/4, for the first toss you bet £0.25, a... | <p>You are given a unique investment opportunity.</p>
<p>Starting with £1 of capital, you can choose a fixed proportion, <var>f</var>, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.</p>
<p>Your return is double your bet for heads and you lose your bet for tails.</p>
<p>For example, if <var>f</va... | 0.999992836187 | Friday, 4th December 2009, 01:00 pm | 3736 | 50% | medium |
348 | Sum of a Square and a Cube | Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.
Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in exactly $4$ different ways.
For example, $5229225$ is a palindromic number and it can be expressed in ex... | Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.
Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in exactly $4$ different ways.
For example, $5229225$ is a palindromic number and it can be expressed in ex... | <p>Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.</p>
<p>Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in <b>exactly</b> $4$ different ways.<br/>
For example, $5229225$ is a palindromic number and it ... | 1004195061 | Saturday, 3rd September 2011, 04:00 pm | 3369 | 25% | easy |
191 | Prize Strings | A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.
During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (abse... | A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.
During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (abse... | <p>A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.</p>
<p>During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and... | 1918080160 | Saturday, 26th April 2008, 01:00 am | 7785 | 35% | medium |
795 | Alternating GCD Sum | For a positive integer $n$, the function $g(n)$ is defined as
$$\displaystyle g(n)=\sum_{i=1}^{n} (-1)^i \gcd \left(n,i^2\right)$$
For example, $g(4) = -\gcd \left(4,1^2\right) + \gcd \left(4,2^2\right) - \gcd \left(4,3^2\right) + \gcd \left(4,4^2\right) = -1+4-1+4=6$.
You are also given $g(1234)=1233$.
Let $\displ... | For a positive integer $n$, the function $g(n)$ is defined as
$$\displaystyle g(n)=\sum_{i=1}^{n} (-1)^i \gcd \left(n,i^2\right)$$
For example, $g(4) = -\gcd \left(4,1^2\right) + \gcd \left(4,2^2\right) - \gcd \left(4,3^2\right) + \gcd \left(4,4^2\right) = -1+4-1+4=6$.
You are also given $g(1234)=1233$.
Let $\displ... | <p>
For a positive integer $n$, the function $g(n)$ is defined as
</p>
$$\displaystyle g(n)=\sum_{i=1}^{n} (-1)^i \gcd \left(n,i^2\right)$$
<p>
For example, $g(4) = -\gcd \left(4,1^2\right) + \gcd \left(4,2^2\right) - \gcd \left(4,3^2\right) + \gcd \left(4,4^2\right) = -1+4-1+4=6$.<br>
You are also given $g(1234)=1233$... | 955892601606483 | Saturday, 23rd April 2022, 02:00 pm | 391 | 30% | easy |
778 | Freshman's Product | If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the freshman's product of $a$ and $b$, denoted $a\boxtimes b$, is the integer $c$ with decimal representation $c=(\dots c_2c_1c_0)$ such that $c_i$ is the last digit of $a_i\cdot b_i$.
F... | If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the freshman's product of $a$ and $b$, denoted $a\boxtimes b$, is the integer $c$ with decimal representation $c=(\dots c_2c_1c_0)$ such that $c_i$ is the last digit of $a_i\cdot b_i$.
F... | <p>
If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the <i>freshman's product</i> of $a$ and $b$, denoted $a\boxtimes b$, is the integer $c$ with decimal representation $c=(\dots c_2c_1c_0)$ such that $c_i$ is the last digit of $a_i\c... | 146133880 | Sunday, 26th December 2021, 10:00 am | 451 | 30% | easy |
40 | Champernowne's Constant | An irrational decimal fraction is created by concatenating the positive integers:
$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$
It can be seen that the $12$th digit of the fractional part is $1$.
If $d_n$ represents the $n$th digit of the fractional part, find the value of the following expression.... | An irrational decimal fraction is created by concatenating the positive integers:
$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$
It can be seen that the $12$th digit of the fractional part is $1$.
If $d_n$ represents the $n$th digit of the fractional part, find the value of the following expression.... | <p>An irrational decimal fraction is created by concatenating the positive integers:
$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$</p>
<p>It can be seen that the $12$<sup>th</sup> digit of the fractional part is $1$.</p>
<p>If $d_n$ represents the $n$<sup>th</sup> digit of the fractional part, find... | 210 | Friday, 28th March 2003, 06:00 pm | 87104 | 5% | easy |
340 | Crazy Function | For fixed integers $a, b, c$, define the crazy function $F(n)$ as follows:
$F(n) = n - c$ for all $n \gt b$
$F(n) = F(a + F(a + F(a + F(a + n))))$ for all $n \le b$.
Also, define $S(a, b, c) = \sum \limits_{n = 0}^b F(n)$.
For example, if $a = 50$, $b = 2000$ and $c = 40$, then $F(0) = 3240$ and $F(2000) = 2040$.
Als... | For fixed integers $a, b, c$, define the crazy function $F(n)$ as follows:
$F(n) = n - c$ for all $n \gt b$
$F(n) = F(a + F(a + F(a + F(a + n))))$ for all $n \le b$.
Also, define $S(a, b, c) = \sum \limits_{n = 0}^b F(n)$.
For example, if $a = 50$, $b = 2000$ and $c = 40$, then $F(0) = 3240$ and $F(2000) = 2040$.
Als... | <p>
For fixed integers $a, b, c$, define the <dfn>crazy function</dfn> $F(n)$ as follows:<br/>
$F(n) = n - c$ for all $n \gt b$<br/>
$F(n) = F(a + F(a + F(a + F(a + n))))$ for all $n \le b$.
</p>
<p>Also, define $S(a, b, c) = \sum \limits_{n = 0}^b F(n)$.</p>
<p>
For example, if $a = 50$, $b = 2000$ and $c = 40$, then ... | 291504964 | Sunday, 29th May 2011, 07:00 am | 1239 | 30% | easy |
189 | Tri-colouring a Triangular Grid | Consider the following configuration of $64$ triangles:
We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edg... | Consider the following configuration of $64$ triangles:
We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edg... | <p>Consider the following configuration of $64$ triangles:</p>
<div class="center"><img alt="" class="dark_img" src="resources/images/0189_grid.gif?1678992055"/></div>
<p>We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same co... | 10834893628237824 | Friday, 11th April 2008, 05:00 pm | 2332 | 70% | hard |
404 | Crisscross Ellipses | $E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$.
$E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around the origin $O(0, 0)$ for $0^\circ \lt \theta \lt 90^\circ$.
$b$ is the distance to the origin of the two intersection points closest to the origin and $c$ is t... | $E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$.
$E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around the origin $O(0, 0)$ for $0^\circ \lt \theta \lt 90^\circ$.
$b$ is the distance to the origin of the two intersection points closest to the origin and $c$ is t... | <p>
$E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$.<br/>
$E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around the origin $O(0, 0)$ for $0^\circ \lt \theta \lt 90^\circ$.
</p>
<div align="center">
<img alt="0404_c_ellipse.gif" src="resources/images/0404_c_ellipse.g... | 1199215615081353 | Sunday, 2nd December 2012, 01:00 am | 359 | 60% | hard |
526 | Largest Prime Factors of Consecutive Numbers | Let $f(n)$ be the largest prime factor of $n$.
Let $g(n) = f(n) + f(n + 1) + f(n + 2) + f(n + 3) + f(n + 4) + f(n + 5) + f(n + 6) + f(n + 7) + f(n + 8)$, the sum of the largest prime factor of each of nine consecutive numbers starting with $n$.
Let $h(n)$ be the maximum value of $g(k)$ for $2 \le k \le n$.
You are give... | Let $f(n)$ be the largest prime factor of $n$.
Let $g(n) = f(n) + f(n + 1) + f(n + 2) + f(n + 3) + f(n + 4) + f(n + 5) + f(n + 6) + f(n + 7) + f(n + 8)$, the sum of the largest prime factor of each of nine consecutive numbers starting with $n$.
Let $h(n)$ be the maximum value of $g(k)$ for $2 \le k \le n$.
You are give... | <p>Let $f(n)$ be the largest prime factor of $n$.</p>
<p>Let $g(n) = f(n) + f(n + 1) + f(n + 2) + f(n + 3) + f(n + 4) + f(n + 5) + f(n + 6) + f(n + 7) + f(n + 8)$, the sum of the largest prime factor of each of nine consecutive numbers starting with $n$.</p>
<p>Let $h(n)$ be the maximum value of $g(k)$ for $2 \le k \le... | 49601160286750947 | Saturday, 19th September 2015, 01:00 pm | 348 | 60% | hard |
190 | Maximising a Weighted Product | Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which $P_m = x_1 \cdot x_2^2 \cdot \cdots \cdot x_m^m$ is maximised.
For example, it can be verified that $\lfloor P_{10}\rfloor = 4112$ ($\lfloor \, \rfloor$ is the integer part function).
Find $\sum\l... | Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which $P_m = x_1 \cdot x_2^2 \cdot \cdots \cdot x_m^m$ is maximised.
For example, it can be verified that $\lfloor P_{10}\rfloor = 4112$ ($\lfloor \, \rfloor$ is the integer part function).
Find $\sum\l... | <p>Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which $P_m = x_1 \cdot x_2^2 \cdot \cdots \cdot x_m^m$ is maximised.</p>
<p>For example, it can be verified that $\lfloor P_{10}\rfloor = 4112$ ($\lfloor \, \rfloor$ is the integer part function).</p... | 371048281 | Friday, 18th April 2008, 10:00 pm | 4583 | 50% | medium |
88 | Product-sum Numbers | A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.
For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.
For a given set of si... | A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.
For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.
For a given set of si... | <p>A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.</p>
<p>For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.</p>
<p>For ... | 7587457 | Friday, 4th February 2005, 06:00 pm | 11591 | 40% | medium |
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