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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Weighted Activity Selection Algorithm
Maximilian113   1
N 2 minutes ago by Maximilian113
An interesting problem:

There are $n$ events $E_1, E_2, \cdots, E_n$ that are each continuous and last on a certain time interval. Each event has a weight $w_i.$ However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:

For each $E_i$ consider $x_i,$ the sum of $w_j$ over all $j$ such that $E_j$ and $E_i$ are not compatible.
1. At each step, delete the event that has the maximal $x_i.$ If there are multiple such events, delete the event with the minimal weight.
2. Update all $x_i$
3. Repeat until all $x_i$ are $0.$
1 reply
Maximilian113
Today at 12:30 AM
Maximilian113
2 minutes ago
J
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Mock 22nd Thailand TMO P3
korncrazy   1
N 12 minutes ago by YaoAOPS
Source: own
Find all triples of positive integers $(a,b,c)$ such that $$a|b+c,\,b|c+a, c|a+b$$and $\gcd(a,b,c)=1$.
1 reply
korncrazy
an hour ago
YaoAOPS
12 minutes ago
J
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Mock 22nd Thailand TMO P2
korncrazy   1
N 14 minutes ago by YaoAOPS
Source: Own
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $B'$ be the reflection of $B$ across the line $AC$ and $B'$ be the reflection of $C$ across the line $AB$. Let $B'C$ and $BC'$ intersect at $A'$. Prove that the orthocenter of triangle $ABC$ coincides with the circumcenter of triangle $A'B'C'$.
1 reply
korncrazy
an hour ago
YaoAOPS
14 minutes ago
J
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Mock 22nd Thailand TMO P1
korncrazy   1
N 20 minutes ago by YaoAOPS
Source: Own, Folklore
Let $a,b,c$ be real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=2$. Find the largest possible value of $abc$.
1 reply
korncrazy
an hour ago
YaoAOPS
20 minutes ago
J
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Index of Coincidence of a ciphertext with respect to itself.
fortenforge   0
Oct 18, 2009
If we are comparing a text to itself, we basically are mathematically finding the probability that if we choose $ 2$ characters from the text, the characters will be the same.
Here is the formula:
$ \displaystyle\sum_{i=1}^{c}\frac{n_i(n_i - 1)}{N(N-1)}$.
where $ c$ is the number of characters in the alphabet, $ n_i$ is the number of times the $ i$th of the alphabet appears in the plaintext, and $ N$ is the number of letters in the plaintext.
Let us try to derive this formula. Probability is defined as the number of ways you get what you want divided by the total number of possibilities. How many ways are there to choose any $ 2$ letters from a group of $ N$ letters? It is of course, $ \dbinom{N}{2}$ which is equal to $ \frac{N!}{2!(N-2)!} = \frac{N(N-1)}{2}$. This is the denominator. To calculate the numerator, we first calculate the number of ways to pick $ 2$ a's from our plaintext and add that to the number of ways to pick $ 2$ b's from our plaintext, and so on. If the number of a's in our plaintext was $ n_i$, then the number of ways to pick $ 2$ a's is $ \dbinom{n_i}{2}$, this is equal to $ \frac{n_i(n_i-1)}{2}$ as we have shown before. This numerator and denominator gives us $ \displaystyle\sum_{i=1}^{c}\frac{n_i(n_i - 1)/2}{N(N-1)/2}$, the $ /2$'s cancel giving us our desired formula:

$ \boxed{\displaystyle\sum_{i=1}^{c}\frac{n_i(n_i - 1)}{N(N-1)}}$.
0 replies
fortenforge
Oct 18, 2009
0 replies
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Why frequency analysis does not work
fortenforge   0
Aug 30, 2009
Frequency Analysis works because there is a one to one correspondence between the plaintext alphabet and the ciphertext alphabet. If frequency analysis is going to work, the letter $ p$ should ALWAYS be encrypted as the letter $ c$. In a Vigenere cipher this does not occur. Depending of $ p$'s position in the plaintext, $ p$ could be encrypted as one of several letters. If the keyword has length $ 5$, then $ p$ could be encrypted as $ c_1,c_2,c_3,c_4,c_5$. This is not a one to one correspondence so frequency analysis does not work.

Let us say that the frequency of $ p$ in normal English was $ i$. If the key word was of length $ 1$, the frequency of $ c$ in the cipher text would be $ i$ as well. But if the key word was of length $ 2$, then the frequency of $ c_1 = i/2$ and the frequency of $ c_2 = i/2$. Basically frequency analysis works if there is one alphabet that corresponds to another alphabet in a 1 to 1 correspondence.
To find a method for cryptanalysis we need to be more creative.
0 replies
fortenforge
Aug 30, 2009
0 replies
J
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Mathematics of the Vigenere Cipher
fortenforge   0
Aug 21, 2009
Ok, so I lied. I said that the next post was going to be about why frequency analysis fails on the Vigenere cipher but I decided to talk about how to mathematically define the cipher.

Let us say that we have already translated our plaintext into numbers (A = 0, B = 1, C = 2, ...). Let us say that the numbers are $ p_0, p_1, p_2, \cdots p_n$.

Let us say that we have chosen a key of length $ x$ and the letters of our key transformed into numbers are $ k_0, k_1, \cdots k_x$.

To encrypt plaintext number $ p_i$ we use $ k_j$ where $ j \equiv i \pmod{x}$. This accounts for the fact that the key is repeated over each letter of the plaintext. We use mod $ x$ because $ x$ is the length of the keyword.

Call $ c_i$ the corresponding ciphertext number to $ p_i$.

$ c_i \equiv p_i + k_j \pmod{26}$ where $ k_j \equiv i \pmod{x}$.

The first part is just the Caesar Cipher mathematically. The only difference is that as $ p_i$ changes $ k_j$ changes as well. This is what makes the Vigenere cipher a much better code.

Let's take an example:

plaintext: BLITZKRIEG
Numerical equivalent: 1 11 8 19 25 10 17 8 4 6

keyword: WAR
Numerical equivalent: 22 0 17

For the $ 0$th number of our plaintext, $ 1$, to find the equivalent key letter we take $ 0 \pmod{3} = 0$. So we take the $ 0$th keyword number which is $ 22$.

For the $ 5$th number of our plaintext, $ 10$ to find the equivalent key letter we take $ 5 \pmod{3} = 2$. So we take the $ 2$nd keyword number which is $ 17$.

We would continue this process for all the letters there by finding each letters keyword equivalent.

Notice that the first number in our list is being considered as our 0th number and the 2nd number is being considered as the 1st number to make the math work. In cryptography this is normal.

You can see that this method works by verifying it here:

WARWARWARW
BLITZKRIEG

The math method and the visual method match up, the 0th plaintext letter corresponds to the 0th key letter and the 5th plaintext letter corresponds to the 2nd key letter.

If we wanted to encrypt the 0th letter mathematically we would find:

$ 1 + 22 \pmod{26} \equiv 23$. So our 0th ciphertext number is 23.

If we wanted to encrypt the 5th letter mathematically we would find:

$ 10 + 17 \pmod{26} \equiv 1$. So our 5th ciphertext number is 1.

We would continue the process for all the letters.

Again when doing it mathematically and doing it without math you get the same ciphertext:

XLZPZBNIVC.

Learning what a cipher is mathematically is not much useful if you are decrypting a message by hand, but it is enormously useful if you are trying to program a cipher on a computer.
0 replies
fortenforge
Aug 21, 2009
0 replies
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Number of Possible Keys for Substitution cipher.
fortenforge   0
Jul 6, 2009
We know that the Monoalphabetic Substitution Cipher should have a lot more keys than the Caesar Cipher, but how many more?

Ptext: ABCDEFGHIJKLMNOPQRSTUVWXYZ
Ctext: ??????????????????????????

Let's look at the first "?" under the "A" in the ciphertext.
How many choices do we have for that "?". Well, we have $ 26$ choices because we can choose any letter of the alphabet. Let us say we chose "R".

Ptext: ABCDEFGHIJKLMNOPQRSTUVWXYZ
Ctext: R?????????????????????????

How many choices do we have for the next "?". We can choose any letter of the alphabet except "R", because we have already chosen that for the plaintext letter "A". So we have $ 25$ choices. Let's say we chose "E". Now for the next question mark we can't chose the letter "R" or "E" so we have $ 24$ choices. By now you should see the pattern, we have $ 26 \cdot 25 \cdot 24 \cdot 23 \ldots$ choices. That is equivalent to $ 26$ factorial.

$ 26! \approx 400000000000000000000000000$.

This is a lot of keys. Much much more keys than a Caesar Cipher. Unfortunately, with today's computers this is not that many keys. But this makes it impossible to try to crack a monoalphabetic substitution cipher using brute force by hand. There is however another method to crack this code...
0 replies
fortenforge
Jul 6, 2009
0 replies
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Mathematics of the Caesar Cipher
fortenforge   0
Jun 21, 2009
We can write the algorithm for the Caesar cipher in terms of math.

$ k$ is the key. $ p$ is the letter being encrypted and $ c$ is the encrypted letter. The variables p and c are used to represent the letter being encrypted because in cryptography we refer to the original message as the 'plaintext' and the encrypted message as the 'ciphertext'.

We know $ k$, represents a number because it is the key. But $ p$ and $ c$ are actually letters. We need to convert them into numbers. This is very simple. Represent A by 0, B by 1, C by 2 ... Z by 25.

When encrypting a message we are shifting it by $ k$ letters. In terms of numbers we are just adding $ k$ to $ p$ to get $ c$.

$ c = p + k$.

There is one problem with this. If $ p = 25$ and $ k = 1$ then $ c = 26$ which is a number we cannot convert to a letter. This problem occurs because of the 'wrapping around' from Z to A. To fix this we can use modular arithmetic. If you don't know what this is try googling it. We will almost always be working in mod 26 because there are 26 letters in the alphabet. Our new equation would be:

$ c \equiv p + k \text{ }(\text{mod } 26)$

This is how to encrypt a message. To decrypt a message instead of adding $ k$ we should subtract it.

$ c\equiv p - k \text{ }(\text{mod } 26)$
0 replies
fortenforge
Jun 21, 2009
0 replies
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TM passes through circumcenter of trapezoid ABCD if M is interior to ABD
parmenides51   0
Jun 27, 2021
Source: 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p8
The trapezoid $ABCD$ ($BC \perp AD$) is inscribed in a circle. On the diagonal $BD$ is randomly selected point $E$. The line $CE$ intersects the side $AB$ at point $K$. Denote by $\omega$ the circumscribed circle around the triangle $CED$. The perpendicular drawn from point $C$ on $AD$ intersects $\omega$ for second time at point $N$, and the circumcircle of triangle $BKE$ intersects $\omega$ for second time at point $M$. Prove that the line $TM$ passes through the center of the circumscribed circle of the trapezoid $ABCD$, if the point $M$ lies inside the triangle $ABD$.

(D. Hilko)
0 replies
parmenides51
Jun 27, 2021
0 replies
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TM passes through circumcenter of trapezoid ABCD if M is interior to ABD
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Source: 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p8
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parmenides51
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parmenides51
#1 h  Jun 27, 2021, 8:37 PM • 1 Y
Y by Mango247
The trapezoid $ABCD$ ($BC \perp AD$) is inscribed in a circle. On the diagonal $BD$ is randomly selected point $E$. The line $CE$ intersects the side $AB$ at point $K$. Denote by $\omega$ the circumscribed circle around the triangle $CED$. The perpendicular drawn from point $C$ on $AD$ intersects $\omega$ for second time at point $N$, and the circumcircle of triangle $BKE$ intersects $\omega$ for second time at point $M$. Prove that the line $TM$ passes through the center of the circumscribed circle of the trapezoid $ABCD$, if the point $M$ lies inside the triangle $ABD$.

(D. Hilko)
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