Found problems: 85335
Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.
Find the sum of the four least positive integers each of whose digits add to $12$.
Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.
Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?
A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square.
Such a board is called [i] interesting[/i] if the following conditions hold:
$\circ$ In all unit squares below the main diagonal, the number $0$ is written;
$\circ$ Positive integers are written in all other unit squares.
$\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers
are actually the numbers $1,2,...,2n$ (not necessarily in that order).
$a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board.
$b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.
A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points.
[i]Proposed by Evan Chen[/i]
Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$.
a) Prove that $AF \perp FC$.
b) Determine the measure of the angle $AFB$.
We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.
An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of
\[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\]
How many guesses are needed for Banana to figure out the quadruple Ana chose?
Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if
\[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
Let $ a_1\equal{}a_2\equal{}1$ and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.
There are $N{}$ friends and a round pizza. It is allowed to make no more than $100{}$ straight cuts without shifting the slices until all cuts are done; then the resulting slices are distributed among all the friends so that each of them gets a share off pizza having the same total area. Is there a cutting which gives the above result if a) $N=201$ and b) $N=400$?
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always 10 cm. Each side of the table is more than 1 meter long. At the initial moment both ants are on the same side of the table.
(a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?
(b) [i](3 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have
$W=(W\cap W_1)\oplus(W\cap W_2)$?
On the unit circle $k$ with center $O$, points $A$ and $B$ with $AB=1$ are chosen and unit circles $k_1$ and $k_2$ with centers $A$ and $B$ are drawn. A sequence of circles $(l_n)$ is defined as follows: circle $l_1$ is tangent to $k$ internally at $D_1$ and to $k_1,k_2$ externally, and for $n>1$ circle $l_n$ is tangent to $k_1$ and $k_2$ and to $l_{n-1}$ at $D_n$. For each $n$, compute $d_n=OD_n$ and the radius $r_n$ of $l_n$.
Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$?
[i]2018 CCA Math Bonanza Individual Round #14[/i]
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
Peter buys a lottery ticket on which he enters an $n$-digit number, none of the digits being $0$. On the draw date, the lottery administrators will reveal an $n\times n$ table, each cell containing one of the digits from $1$ to $9$. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.
A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be considered [i]corner-odd[/i]; i.e. no $1\times k$ rectangle can be [i]corner-odd[/i] for any positive integer $k$.
The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ .
Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.
Several $good$ points, several $bad$ points and several segments are drawn in the plane. Each segment connects a $good$ point and a $bad$ one; at most $100$ segments begin at each point. We have paint of $200$ colors. One half of each segment is painted with one of these colors, and the other half with another one. Is it always possible to do it so that every two segments with common end are painted with four different colors?
[i](M. Qi, X. Zhang)[/i]
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.
Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$)
[i]Proposed by Wong Jer Ren[/i]