Found problems: 85335
For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\} $, let $P_S$ be the product of all differences between two of the elements, namely
$$P_S=(a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$
Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
Evaluate
\[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]
In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board [i]can be attacked[/i] by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less.
In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.
Answer either (i) or (ii).
(i) In a right prism with triangular base, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base), show that these faces are of equal area and perpendicular to each other when the volume attains its maximum.
(ii) Show that \[ \frac{\frac x1 + \frac {x^3} {1 \cdot 3} + \frac {x^5} {1 \cdot 3 \cdot 5} + \frac {x^7} {1 \cdot 3 \cdot 5 \cdot 7} + \cdots }{1 + \frac {x^2} 2 + \frac {x^4}{2 \cdot 4} + \frac{x^6}{2 \cdot 4 \cdot 6} + \cdots} = \int_0^x e^{-t^2} \mathrm dt.\]
Prove that
$$\frac{2}{2!}+\frac{7}{3!}+\frac{14}{4!}+\frac{23}{5!}+...+\frac{k^2-2}{k!}+...+\frac{9998}{100!}<3$$
where $n! = 1 \times 2 \times ... \times n.$
(V Senderov)
$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
A ‘magic square’ of size \(n\) is an \(n\times n\) array of real numbers such that all the rows, all the columns and the two main diagonals have the same sum. Determine the dimension, over \(\mathbb{R}\), of the vector space of \(n\times n\) magic squares.\\
Consider all $1001$-element subsets of the set $\{1,2,3,...,2015\}$. From each such subset choose the median. Find the arithmetic mean of all these medians.
[i]Proposed by Michael Ren[/i]
Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$.
[b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$
[b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.
Find the remainder when $$\sum_{x=1}^{2024} \sum_{y=1}^{2024} (xy)$$ is divided by $31.$
In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC, AF \perp BC$, and $BD = DC = FC = 1$. Find $AC$.
A. $\sqrt{2}$
B. $\sqrt{3}$
C. $\sqrt[3] {2}$
D. $\sqrt[3] {3}$
E. $\sqrt[4] {3}$
The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is?
(A)$89$
(B)$179$
(C)$168$
(D)$119$
Compute
\[
\left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right).
\]
[i]2020 CCA Math Bonanza Team Round #4[/i]
Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$.
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$. We are also given that $\angle ABC = \angle CDA = 90^o$. Determine the length of the diagonal $BD$.
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule:
(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.)
(b) Otherwise, $L_i$ is [i]on[/i] right now.
Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.
Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$.
a) Prove that $\vartriangle PQH \sim \vartriangle AKV$.
b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide]
[hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide]
[img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img]
[url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]
A complete set of the Encyclopedia of Mathematics has $10$ volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
Barb the baker creates a new temperature system for baking bread, Breadus, which is linearly based on Fahrenheit. Bread rises at $110$ F$^\circ$, which is $0$ on the Breadus scale. Bread bakes at $350$ F$^\circ$, which is $100$ on the Breadus scale. Bread is done when it’s internal temperature is $200$ F$^\circ.$ What is this temperature on the Breadus scale?
$\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$
Let $p$ be a prime of the form $p=4n-1$ where $n$ is a positive integer. Prove that
$$\prod_{k=1}^p(k^2+1)\equiv4\pmod p.$$