Found problems: 85335
If $f(x)=x^4+4x^3+7x^2+6x+2022$, compute $f'(3)$.
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 46$
Let $x$, $y$, and $z$ be real numbers satisfying the system of equations
\begin{align*}
xy+4z&=60\\
yz+4x&=60\\
zx+4y&=60.
\end{align*}
Let $S$ be the set of possible values of $x$. Find the sum of the squares of the elements of $S$.
Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities
$$ 4016016\le x<y<z\le 4020025. $$
If $\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}$, then the range value of $\frac{\alpha}{3}$ is
$\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}$
$\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}$
$\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$
$\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\]
Kermit writes down the numbers $1$, $2$, $3$, $4$, $5$. He then erases one number, and discovers that the sum of the remaining numbers is $13$. Which number was erased?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.
A convex quadrilateral $ABCD$ has an inscribed circle touching its sides $AB$, $BC$, $CD$, $DA$ at the points $M$,$N$,$P$,$K$, respectively. Let $O$ be the center of the inscribed circle, the area of the quadrilateral $MNPK$ is equal to $8$. Prove the inequality $$2S \le OA \cdot OC+ OB \cdot OD.$$
The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.
There is a bag with balls whose colors are $c_1, c_2, \dots, c_n$. Let $a_i$ be the number of balls inside the bag with color $c_i$. We are drawing $n$ balls from the bag one by one with replacement. If $p(a_1,a_2,\dots, a_n)$ denotes the probability that at least two of them have same color, which one below is smaller?
$\textbf{(A)}\ p(2,2,2,1) \qquad\textbf{(B)}\ p(1,1,1,1) \qquad\textbf{(C)}\ p(2,2,3) \qquad\textbf{(D)}\ p(2,2,1) \qquad\textbf{(E)}\ p(1,1,1)$
At time $0$, an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?
There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.
Each of the thieves in the $n$-member party ($n \ge 3$) charged a certain number of coins. All the coins were $100n$. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.
Let $x_1, x_2, x_3, y_1, y_2, y_3$ be real numbers in $[-1, 1]$. Find the maximum value of
\[(x_1y_2-x_2y_1)(x_2y_3-x_3y_2)(x_3y_1-x_1y_3).\]
Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$
Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.
If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.
Consider two circles $\Pi_1$ and $\Pi_1$ tangent externally at point $S$, such that the radius of $\Pi_2$ is triple the radius of $\Pi_1$. Let $\ell$ be a line that is tangent to $\Pi_1$ at point $ P$ and tangent to $\Pi_2$ at point $Q$, with $P$ and $Q$ different from $S$. Let $T$ be a point at $\Pi_2$, such that the segment $TQ$ is diameter of $\Pi_2$ and let point $R$ be the intersection of the bisector of $\angle SQT$ with $ST$. Prove that $QR = RT$.
An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths.
(1) Determine the minimum good number.
(2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)
A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$
Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.