Found problems: 85335
$4(299)+3(299)+2(299)+298=$
$\text{(A)}\ 2889 \qquad \text{(B)}\ 2989 \qquad \text{(C)}\ 2991 \qquad \text{(D)}\ 2999 \qquad \text{(E)}\ 3009$
Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that [b]a)[/b]$Q_n$, [b]b)[/b] $K_n$ or [b]c)[/b] $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?
Prove that every polynomial is a difference of two increasing polynomials.
Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\Gamma$ with $AB$ as diameter. Let $E$ be the intersection of the diagonals $AC$ and $BD$. The tangents to $\Gamma$ at the points $C,D$ meet at $P$. Prove that $PC=PE$.
Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .
Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove:
[b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $
[b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $
Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.
What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
[i]Proposed by A. Golovanov[/i]
Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.
Numbers $d(n,m)$, with $m, n$ integers, $0 \leq m \leq n$, are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and
\[md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.\]
Prove that all the $d(n,m)$ are integers.
Find the remainder when $2020!$ is divided by $2020^2.$
[i]Proposed by Kevin Zhao[/i]
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions
(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.
(ii) $F(1- x) = 1 - F(x)$.
Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .
In quadrilateral $ABCD$, angles $A, B, C, D$ form an increasing arithmetic sequence. Also, $\angle ACB = 90^o$ . If $CD = 14$ and the length of the altitude from $C$ to $AB$ is $9$, compute the area of $ABCD$.
The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.
Betty used a calculator to find the product $ 0.075\times 2.56 $. She forgot to enter the decimal points. The calculator showed $ 19200 $. If Betty had entered the decimal points correctly, the answer would have been
$ \text{(A)}\ .0192\qquad\text{(B)}\ .192\qquad\text{(C)}\ 1.92\qquad\text{(D)}\ 19.2\qquad\text{(E)}\ 192 $
Patty has $ 20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $ 70$ cents more. How much are her coins worth?
$ \textbf{(A)}\ \$1.15\qquad
\textbf{(B)}\ \$1.20\qquad
\textbf{(C)}\ \$1.25\qquad
\textbf{(D)}\ \$1.30\qquad
\textbf{(E)}\ \$1.35$
Let $\mathcal{S}$ be a set of $10$ points in a plane that lie within a disk of radius $1$ billion. Define a $move$ as picking a point $P \in \mathcal{S}$ and reflecting it across $\mathcal{S}$'s centroid. Does there always exist a sequence of at most $1500$ moves after which all points of $\mathcal{S}$ are contained in a disk of radius $10$?
[i]Advaith Avadhanam[/i]
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.
[i]Author: Alex Zhu[/i]
Two circles, $\omega_1$ and $\omega_2$, centred at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meets $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1, O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.
Find the next number in the sequence $131, 111311, 311321, 1321131211,\cdots$
Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.