Found problems: 85335
Let $P(x)$ be the polynomial of degree at most $6$ which satisfies $P(k)=k!$ for $k=0,1,2,3,4,5,6$. Compute the value of $P(7)$.
A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$.
a) Find the total number of cute numbers.
b) Prove that the sum of all cute numbers is divisibel by $1408$.
Let $f$ be a diff erentiable real-valued function defi ned on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?
Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.
Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.
For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called [i]beautiful[/i] if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$).
a. For $n=6$, point out a [i]beautiful [/i] permutation.
b. Prove that there exists a [i]beautiful[/i] permutation for every $n$.
The function f has the following properties :
$f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$
$f(4) = 10$
Calculate $f(2001)$.
If $0<\alpha<\beta<\gamma<2\pi$, for all $x\in\mathbb{R}$, $\cos(x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0$, then $\gamma-\alpha=$________.
A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar reassignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. Points $A_1$, $B_1$, and $C_1$ are chosen on its incircle. Compute the maximum possible sum of the areas of triangles $A_1BC$, $AB_1C$, and $ABC_1$.
[i]2020 CCA Math Bonanza Lightning Round #1.4[/i]
Suppose we have a cubic polynomial $p(x)$ such that $p(0)=0,p(1)=1,$ and $p(x)\leq \sqrt x$ for $0\leq x \leq 1.$ Suppose $p(0.5)$ is maximized. What is the sum of $p(0.25)+p(0.75)?$
[i]Proposed by Ishin Shah[/i]
How many ways are there to make two $3$-digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$, $2$, $3$, $6$, $7$, $8$ are used exactly once?
Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?
Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.
Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters.
[i]Proposed by Richard Chen[/i]
Yao Ming is $7\text{ ft and }5\text{ in}$ tall. His basketball hoop is $10$ feet from the ground. Given that there are
$12$ inches in a foot, how many inches must Yao jump to touch the hoop with his head?
Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.
The wording is just ever so slightly different, however the problem is identical.
Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
Shayan is playing a game by himself. He picks [b]relatively prime[/b] integers $a$ and $b$ such that $1<a<b<2020$. He wins if every integer $m\geq\frac{ab}{2}$ can be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$. He hasn't been winning often, so he decides to write down all winning pairs $(a,b)$, from $(a_1,b_1)$ to $(a_n,b_n)$. What is $b_1+b_2+\ldots+b_n$?
[i]2020 CCA Math Bonanza Tiebreaker Round #2[/i]
Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.
There are five cities $A,B,C,D,E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes $A \to B \to C \to A$ and $A\to C \to B \to A$ are different.)
In the game of [i]Winners Make Zeros[/i], a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
[list]
[*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$.
[*] If $m<n$, the player choose a positive integer $c$ such that $n-cm\geq 0$, and replaces $(m,n)$ with $(m,n-cm)$.
[/list]
When $m$ or $n$ becomes $0$, the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$, find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move.