Problem 4 (conditional probability): Bayes’ Theorem states that It is easy to see this, since if you move $P(B)$ to the other side, then both sides are equal to $P(A \cap B)$ Problem 6 (stars and bars): If you have $n$ elements and $k$ bars, then you have $\binom{n+1}{k}$ ways…
Problem 16 (3D Geometry): 3-4-5 triangles are very important! These right triangles will most likely simplify a lot of calculations when used correctly. For this problem, I was unable to solve it because I didn’t use one of the 3-4-5 triangles as the base. Problem 23 (probability): If something is…
Problem 16 (number theory): The number of ways to write any positive integer $N$ as a sum of positive consecutive integers is the number of odd divisors of $N$. Problem 17 (angle bisectors): Angle bisectors can and often apply to more than just one triangle. Problem 18 (graphing): When having…
Problem 12 (angle bisectors): Sometimes, formulas/properties apply on multiple cases. For example, in the following diagram, the angle bisector theorem applies on both $\triangle ABD$ and $\triangle ADF$. Problem 13 (algebra): If something seems to be complicated spend some time looking for an alternative approach first before continuing with it….
Problem 12 (complex): Roots of Unity are commonly found in problems! They aren’t necessarily scaled by $1$. For this problem, I split the points across $0$ to $\pi$ instead of $0$ to $2\pi$. Problem 19 (number theory): Answer choices can help to eliminate four possibilities, leaving you with the answer….
Problem 15 (trigonometry): Always observe the answer options before you start to work on a problem. For example, if the answer options show bounds, then don’t go ahead and try to calculate the actual value. Problem 16 (circles): Descartes’ Circle Theorem states that $2(a^2 + b^2 + c^2 + d^2)=(a+b+c+d)^2$,…
Problem 8 (centroids): Given points $(a,b)$, $(c,d)$, and $(e,f)$ on the coordinate plane, the coordinates of the centroid are $(\frac{a+c+e}{3},\frac{b+d+f}{3})$. Problem 11 (divisibility): READING THE QUESTION MULTIPLE TIMES WON’T HURT. More time would be wasted just guessing around the the problem meant. Problem 13 (centroids): Centroids of a triangle will…
Problem 11 (paper folding): When doing problems about folding something, just draw a perpendicular line across the midpoint and that is the line of reflection. Problem 15 (symmetry): Always consider symmetry, but also consider the edge cases too. Problem 17 (area geometry): Not everything is as it seems! Also, don’t…
Problem 10 (2 x 3 graph theory): Problems rarely make you just brute force everything out. Analyze the situation, come up with anything you can think of that makes progress. Problem 15 (Finding area): Consider your options. Don’t be set on one method or idea for a problem. Problem 17…
The $\texttt{lower_bound}$ function finds the first position where a value could be inserted in without breaking sorter order. Its time complexity is $\texttt{O(log n)}$. For example: $\texttt{it}$ refers to the address of the value. $\texttt{*it}$ refers to the value, and to get the index, use $\texttt{it-v.begin()}$.
Lemma. If $a, b \geq 0$, then $a^3 +b^3 \geq a^2 b + a b ^ 2$. Proof. Since $a,b\geq 0$, $a+b\geq 0$. Also, by the trivial inequality, $(a-b)^2 \geq 0$. Thus, $(a-b)^2 (a+b)\geq 0$. Expanding, $a^3 + b^3 – a^2 b – a b^2 \geq 0$. So, $a^3 +…
$(a-b)^3=a^3-3a^2 b+3a b^2-b^3$ (remember the 3’s) $(a+b)^3=a^3+3a^2 b+3a b^2+b^3$ (remember the 3’s) $1^2 + 2^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}$ $1^3 + 2^3 + … + n^3 = \frac{n^2 (n+1)^2}{4}$ $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ Descartes’ Circle Theorem: For an $m \times n$ grid of numbers satisfying: let $a_{i,j}$ =…
If $p, q$ are distinct natural numbers, and in an arithmetic sequence {${a_n}$}, $S_p = S_q$, where $S_k$ denotes the sum of the first $k$ terms of the sequence {${a_n}$}. Prove that $S_{p+q}=0$. The formula for an arithmetic series is $S_n = n a_1 + \frac{n(n-1)}{2}d$. So, $S_p = p a_1+ \frac{p(p-1)}{2}d$ and $S_q = q a_1+ \frac{q(q-1)}{2}d$. Since $S_p…
Problem: Let point $P$ be outside circle $\Gamma$. The tangents from $P$ to $\Gamma$ hit $\Gamma$ at $A$ and $B$. A third line through $P$ hits $\Gamma$ at $C$ and $D$, such that $C$ is between $P$ and $D$. Point $Q$ is on chord $CD$ such that $\angle DAQ$ =…
In Graph Theory, the diameter of a tree is the largest distance between any two points. The radius is the minimum distance of the maximum distance of any node on the of the diameter to one of the two diameter endpoints. Take the following graph as an example. In this…
Dijkstra’s algorithm is a method to calculate the shortest path from one node to another with weighted edges. It does not work for negative edge weights, however. Fun fact: it is a main algorithm used in Google Maps. It starts off by setting all the distances of each node to…
Here is a random story I did a few years ago when I was bored. The weather forecast had been on the headlines for many days. The citizens of Winnipeg had been warned about the possible storm, but they were ignoring it by thinking about the upcoming Thanksgiving holiday instead….
Watership Down is a novel written by Richard Adams, published in 1972. It tells the story of a group of rabbits and their quest for a new home. The story begins in the British countryside, where a young rabbit named Fiver had a disturbing vision of their warren being destroyed….
Imagine you’re learning something new, like a new language. Have you noticed that after you first learn them, you can remember them quite well, as time goes on, you start forgetting them? This is what the Ebbinghaus forgetting curve is all about. Over a century ago, a psychologist named Hermann…
The story begins with Stanley Yelnats, a young boy from a family cursed with bad luck. He is wrongly accused of stealing a pair of sneakers that once belonged to a famous baseball player and is sent to Camp Green Lake, a juvenile detention facility. Despite the name, Camp Green…
Objects like asteroids and comets in the Solar System can collide with larger planetary bodies, resulting in impact events. When these smaller solar system bodies (SSSBs) strike planets, moons, or other objects, they create a massive explosion similar to a bomb blast. This explosion blows away surface material, leaving behind…
The vastness of space and the universe has always amazed us, sparking our imagination for as long as we could remember. To explore the areas the naked eye can’t see, astronomers use special tools called telescopes. This is their history, how they work, and how they help us understand the…
Cryptocurrency is a special kind of digital money that works without a ledger. Instead, it uses a clever way to keep track of transactions and make them safe. Some popular examples are Ethereum, Tether, BNB, and Dogecoin, but the most popular is Bitcoin. Imagine if everyone who uses the money…
Using brute force isn’t particularly the best solution when it comes to determining primes. Brute Force Example: Sieve of Eratosthenes: This method tries all the numbers, but as soon as a prime number is found all the multiples of it is marked as not prime, meaning there are less options…
Imagine a world without cybersecurity. You open up your bank account to check on your investments – what happened? There is no more money! This is what would happen without cybersecurity, the state of being protected against the criminal or unauthorized use of electronic data. Cyber means anything to…
This program generates a random password. The password is at least 10 characters long and includes at least 1 number and special character. You can tell the program how many special characters, numbers, and letters you want. The password has letters, numbers, and other characters randomly dispersed throughout. For example,…
Copy and paste this code into a python IDE, and you got tic tac toe!
Don’t have a calculator? Don’t worry. Just paste this code into Python and let Python do all the work.
In October 1934, The Long March occurred. It was a series of marches by the Chinese communists, who were led by Mao Zedong. Before the Long March even started, the communists were losing badly to the nationalists. The communists needed to make a plan. They also wanted the nationalists to…
The Silicon Valley Bank Became Bankrupt Silicon Valley Bank, SVB for short, was ranked number 63 in 2021 according to the USA’s top banks. In 2022, it skyrocketed to number 16, an insane feat. In 2023 however, it became bankrupt within a span of 2 days. Here is how this…
Hazel: A young leader of the group of rabbits. Fiver’s brother. Hazel lead the group through predators and conflicts. He is a quick thinker, often putting himself at risk in order to save othes. Fiver: A small rabbit, Hazel’s brother. He is a seer, which means he can predict things…
ctrl+B: make bold ctrl+I: make text slanted crtl+u: underline ctrl+w: close tab ctrl+shift+w: close window ctrl+1: first tab of chrome ctrl+2: second tab of chrome … windows+d: home page ctrl+t: open new tab ctrl+shift+t: open recently closed tab ctrl+n: open new window F11: full screen ctrl+prntscr: unsaved screenshot windows+prntscr: saved…
In python, %d is similar to %s, but uses numbers instead of strings. This prevents typing str() continuously. The %02d could be useful in creating leading 0s. This could be used in digital clocks that need to always have 4 digits. Example: Output: Example: Output:
Soldering is the process of using a soldering iron to melt some metal into liquid. The liquid is placed on something, and when it hardens the things is stuck with a conductor. In soldering, most of the time a printed circuit board (PCB) is used. A PCB is a board…
Rickroll is an internet prank, originated from the song “Never Gonna Give You Up“. Basically, your goal is to avoid that song, or the person Rick Astley moving. The art of Rickroll There is a website called https://www.secretrickroll.com/. This website allows you to create a “link” that leads to rickroll….
In aruduino, servo motors are motors that can rotate a total of 160 degrees. Using gears and something on top of the motor, servo motors can rotate around at mildly fast speeds. The yellow wire goes to a GPIO pin. The red wire goes to 5v. The black wire goes…
Nailed It! is a CCC senior 3/ junior 5 question. The method to solve this is similar to Escape Room. The first thing to do to solve this problem is to create a list that measures that number of boards that are length N into the term N. Then, we…
Remote controll A remote controll uses something that can detect the numbers pressed on a remote controll. It is called the Infrared Receiver. Like in the name, the receiver uses infrared light, which cannot be seen by the naked eye, to detect remote controll signals. Liquid Crystal Display An LCD…
For a Junior 5 question, Escape Room isn’t that easy. The brute-force method would be to use breadth first search and find factors, but that will almost guarantee the time will exceed the limit. To solve this, we will need to use a method that doesn’t require any calculations involving…
Modern Art is a CCC problem. Notice that if you brush on a specific square for an odd number of times, it ends up being a gold square. If it is even or 0, it is a black square. Using this, we could create a table that has the numbers….
An air conditioner uses something called refrigerant. In a hot room, the refrigerant will take away the heat from the room. The heat will evaporate. The more the refrigerant contacts, the more heat it will take away. A compressor is another part of an air conditioner. If the refrigerant is…
4 victims were shot at 6:20 AM. 2 were confirmed dead, while a suspect was shot as well. One man was found dead by a Langley city bus loop. Another man was discovered dead at the Creek Stone Place. The police also found a woman who’d been shot near 203A Street…
Imagine you are given a long-term piece of homework. It is due in six months. You plan to complete some every day, where you finish right before the deadline. 2 months went by, and you haven’t even started working on it. You change everything where you complete everything in 4…
To make an Arduino Light Project, you will need: 1 Arduino board 1 breadboard 15 wires 7 220 ohm resistors 7 Light Emitting Diodes Connect them all where the lights are side-by-side. Then, use the following code: This should create a lot of flashing lights.
Mouse Journey is a problem that is related to simple math. Read the problem first. The logic is simple: the total number of possibilities is the number of possibilities for the one above and the one to the left. The start is 1, and the cat areas are 0. Now,…
Floor plan is a CCC senior 3 problem that is very confusing at first, and gradually becomes more understandable. IndexError is more common. Functions greatly aid to help make the code for the solution. First, make a function called findarea. This function, given a coordinate, will find the area of…
Babbling Brooks is an example of a problem that is good for functions. There could be two functions, one of them seperates and the other joins. There could be a simple part that isn’t in a function, but most of the work and ligic is into the functions. Sample solution…
Knight Hop is a Junior 5 level 7 point problem. A little bit of chess knowledge is very useful, as it requires a knight. Functions are also extremely helpful, especially if it determines the possible next moves given a set of points. First, create a function than takes in a…
https://dmoj.ca/problem/ccc14s3 The problem above is mainly based off logic. It really requires only a few if else statements, variables and loops. First, you need to create a loop that loops based of an integer input. Everything else will be in this loop. Create an empty list that is for the…
https://dmoj.ca/problem/ccc14s1 This is a dmoj 5-point Junior 4 question. It is more complex than it seems. Read on to find the solution and the logic to it! First, read the question. Without knowing what the question is about, all the rest of this blog will be useless. First things first,…
Conjucates are very useful when it comes to simplifying fractions with an irrational denominater. A conjugate is where you change the signs between two terms. Take a look at the polynomial below: If you try to add this fraction to a rational fraction like 1/3, it will be very exhausting…
Passwords are things that can block a hacker from going into your account. If you remember the passwords, only you would be able to access your account. This means that secure passwords are best, otherwise there would be basically no point of having one. Below is some of the most…
Unbound is a novel in verse that is about slavery. It was written by Ann. E. Burg. In the beginning, Grace, a nine-year-old girl, was called to the Big House to do slave work. After a few weeks, she couldn’t stand the pressure anymore, and escaped, taking a big risk….
D-Day has been one of the most important days of war history. It started on June 6, 1944. Without it, it’s likely that World War 2 would have resulted in a German victory. However, this day creates a question that is very important. How did D-Day lead to victory for…
Russia had recently started invading Ukraine. This sudden news had made a lot of people research things they hadn’t really paid too much attention to before. The Russians’ relationship wasn’t good with the Ukrainians’, and held a grudge against them for almost a decade. Russia’s neighbor also wanted to be…
Shang-Chi and the Legend of the Ten Rings is one of the latest movies released by Marvel Studios. The film starts by explaining what the Ten Rings are, and who possessed them, Wenwu. The Ten Rings gave the possessor power and eternal life. Nobody knew how he got the rings….
People sneeze when there are a certain number of irritants in the person’s throat or nose. Many people often sneexe multiple times before stopping, because one sneeze wouldn’t be enough to get rid of all the dust particles. So, another sneeze occures. Reasearchers aren’t exactly sure, but that is what…
In C++, the function atoi() is designed to convert characters to integers. If a character is named n, all that is needed to convert it to an integer is: In other words, the atoi function wraps around the character and changes it to an integer.
Initial value for variables is very important in C++. If a variable is in the wrong scope, errors can appear, or it won’t give the expected outcome. For example, if a variable is declared in a for loop, and it is used in a function, the announced variable will not…
Matrices are among an important group in coding problems. Matrix rotations are also vital. For instance, QR codes can be read even when it is rotated 90°. Take the coding problem below as an example: https://dmoj.ca/problem/ccc18s2 This is a matrix problem. At first glance, people might thnk that the data…
This type of cipher is one of the easiest because each encoded letter could only be another. It is also known as the mirror cipher, because A=Z, B=Y, C=X, and so on. This type is best to do a key, and it is below: Here is a code in Atbash:
This type of cipher is one of the most modern and ancient types. A message is encoded by shifting the alphabet by a certain number of times. Then, the new alphabet is put. For instance, in a shift of 1, A=B, B=C, and so on, all the way to Z=A….
Cryptography is the study of codes, encoding them, and decoding them. It is useful in wars, when sodiers want to send messages to each other. To avoid enimies intercepting and reading the message, senders would decrypt them with some type of format, such as Caesar Cipher. See if you can…
Morphine is a drug used to kill severe pain. Though this drug can save lives, simultaneously it kills people as well. Despite being pain-killers, morphine is still a type of drug. If people have it and they aren’t going through severe/moderate pain, they can be addicted to it and eventually…
In C++, value scope is important for declaring variables. If a variable is declared in a for loop, it is different than if it is announced in the main() function. If the variable is declared in the for loop, it will only be present in the loop. In the main()…
Remember: in C++, a frequent thing that needs to be done is to test (use debugging code) to check that there are no mistakes. Why? Mistakes could be solved easily, but if there are too many all of them would be hard to solve. Plus, if an entire program was…
In C++, the stoi() function is useful to change strings to integers. Basically, it only needs one parameter, and it is like stoi(something). Example: Example Output: And that’s all! In short, stoi() is a function that converts strings into integers.
Arranging Books is a coding problem that can be viewed here. It isn’t easy, and is literally impossible to solve it correctly by actually swapping books. Here is the problem: Canadian Computing Competition: 2021 Stage 1, Junior #4 Valentina wants books on a shelf to be arranged in a particular…
In C++, the substr() method is one of the easiest functions to use to split a string into multiple slices. Here are the parameters of the substr(): where name is the string that you want to be sliced, position is the staring point, and len is the number of characters…
When trying to earn money investing stocks, remember not to borrow money from the bank to buy them! There would be a huge profit if you get money from buying and selling them, but what if the stock goes down and you lose money? You wouldn’t have enought money to…
Protests should be allowed freely to express one’s right and choice. One of the basic reasons is people need freedom. We deserve the right to make our own choices. If people protest against the same issue, they realize that they aren’t alone. By protesting, humans will feel better because they…
Face masks should be mandatory during Covid-19 for many reasons. Fundamentally, this virus spreads mainly be the mouth, like when someone sneezes. However, if something like that happens when the person is wearing a mask, Covid-19 will not be that easy to outspread. Alternatively, masks can protect us in various…
Python is a popular programming language created by Guido van Rossum, and released in 1991. It is used for: software development mathematics What can Python do? Python can connect to database systems. It can also read and change files. Python can be used to handle big data and perform complex…
There are many ways a forest fire is extremely dangerous. One reason is the smoke from the fires can damage the health of people and other living things. Unfortunate enough, the bad air goes to the lungs of people, causing breathing issues and we may die. Secondly, forest fires can…
1: Yes. 2: Examine the last digit of N. If it is an even number, then N is divisible by 2. 3: Add the sum of the digits of N and call it P. If P is divisible by 3, than N is divisible by 3. 4: Take the last…
Herbivores Herbivores are animals that eat only plants, such as grass. For example, cows and sheep are herbivores because they eat grass and hay. Carnivore Carnivores are animals that only eat meat, like rabbits or beef. For instance, a polar bear is a carnivore because they eat seals and fish….
Find out how to attract bees to your garden and bring it to life!
On the top left corner, click file-settings or do ctrl+alt+s. Go to Python Interpreter. Click the plus sign that is near the bottom. Search up the package you want to install, like numpy or matplotlib. Click Install Package Wait until it is installed, and then you are done!
Writing First, an author writes the book and paper. He or she will then submit it to another person to go through editing and publishing. Typing The person that has the book types it in and later gives it to an editor. Editing The Story The editor edits the story…
Fastquant is a new python package that is related to stocks. Using only two lines, you can predict which stocks are good to buy, and which stocks you should sell. First, install the fastquant package. Then, type the following: Run the program, and on the top left you will find…
A Stem and Leaf Plot is a special table where each data value is split into a “stem” (the first digit or digits) and a “leaf” (usually the last digit). For example: For every first digit of a number, it is in the “Stem” part. The other digit is in the “Leaf”…
The class really did love their new hamster, Fuzzy. Everyone wanted to hold her, feed her, put her in the plastic hamster ball and watch her roll around the classroom. They didn’t even mind cleaning her cage. The class was warned that hamsters are very good escape artists. Mrs. Elrod…
There once were two people who lived near by a river. They worked together to get a supply of water by using Chinese water buckets. The two put water in two buckets, put them on a wooden pole and carry the water back to the cabin. They lived happily without…
In mathematics, a parabola is a curve which is mirror-symmetrical and is close to a U-shape. It fits several different mathematical descriptions, which can all be proved to define exactly the same curves.
Stay away from people who are coughing or sneezing Since COVID-19 is an infection, coughing and sneezing are common symptoms. Additionally, coughing and sneezing both release the virus into the air, so they may increase your risk of infection. Keep your distance from people who appear to have symptoms of…
There once was a raccoon village that was stocked with food. The procyon lotors lived peaceful, happy long lives until three raccoons were born. They were taught to take only what they need. However, the kits were greedy, and did the opposite of the leader’s rules. They stole all the…
Gold is a precious metal, so it is often copied in fake jewelry and metal blends. By most standards, anything consisting of less than 41.7%, or 10 karats of gold is considered to be fake. If you’re wondering whether your gold is real, the most reliable test is to take…
You want to try sudoku, but you don’t know where to begin. Sudoku looks hard because it involves numbers, but in reality, it’s not math-based. Even if you think you’re bad at math, you can still do well at sudoku. In fact, the numbers could be replaced with letters or…
The Rule of 72 is a handy tool used in finance to estimate the number of years it would take to double a sum of money through interest payments, given a particular interest rate. The rule can also estimate the yearly interest rate required to double a sum of money in a specified number…
Poison ivy is a common plant on the North American continent, noted for its ability to create an itchy rash on contact with skin. It is an extremely adaptable, persistent type of vegetation and as a result, it can be easy to accidentally become affected. Fortunately, it is not all…
When shooting in a soccer match, remember: shoot low! A study found that the percentage of the goals that are scored high are as followed: Top Left: 8 percent Top Center: 4 percent Top Right: 5 percent Yikes. The possibility isn’t high. Middle Left: 7 percent Middle Center: 8 percent Middle Right: 6 percent Although the…
Three Chinese astronauts arrived Thursday at China’s new space station, starting a three-month mission that marked another milestone in the country’s space program. Commander Nie Haisheng, followed by Liu Boming and Tang Hongbo, flew in a rocket called Shenzhou-12, and set off. The rocket dropped its boosters about two minutes later, followed…
This Monday, at the press conference earlier to Portugal’s game against Hungary, he removed two bottles of Coca-Cola from his vicinity. The Portuguese star sat down, saw the two bottles of sugary drinks and moved them away before taking a water bottle. His gesture clearly meant “drink water”. The stock…
It was a day where the sun shone brightly over neighbourhood Oldville. The houses there seemed alike: pale peach walls, dark brown tiles for the roof and very small yards. The gates to the entrance creaked open, and two people sauntered over to start a long day of taking care…
Alzheimer’s disease is the most common type of dementia. It is a progressive disease beginning with small memory loss and leads to loss of the ability to carry on a conversation. Alzheimer’s disease can seriously affect a person’s ability to carry out daily activities. Younger people may get Alzheimer’s disease, but…
To make a flowing water light, you will need the following: 11 wires 1 bar graph LED 1 breadboard 10 220 ohm resistors 1 raspberry pi 3 Now, place the pieces like the following: Once you are done putting the pieces in the right place, type the following code: Once…
The pieces on the rows of the breadboard are connected! If you try to do an LED where the 2 wires are on the same row, the LED won’t light up. Make sure the pieces on the breadboard are in the right position. If the pieces are in the wrong…
