Found problems: 85335
Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT".
[i]Proposed by Alex Li[/i]
Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty.
We can perform any combination of the following operations:
- Pour away the entire amount in bucket $X$,
- Pour the entire amount in bucket $X$ into bucket $Y$,
- Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount.
[b](a)[/b] How can we obtain 10 litres of 30% syrup if $n = 20$?
[b](b)[/b] Determine all possible values of $n$ for which the task in (a) is possible.
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation
\[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\]
\[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\]
\[+(x - b)(x - c)(x - d)(x - e) = 0\]
has four distinct real solutions.
Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$.
Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$
[asy]
size(250);
path p=rotate(45)*polygon(4);
int i;
for(i=0; i<5; i=i+1) {
draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p);
}
label("$S_1$", (0,-0.75));
label("$S_2$", (1,-0.75));
label("$S_3$", (3/2,-0.75));
label("$\cdots$", (7/4, -3/4));
label("$\cdots$", (2.25, 0));[/asy]
A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to $\mathcal{P}.$
[center]
[img]
https://cdn.artofproblemsolving.com/attachments/a/8/7607ee5d199471fd13b09a41a473c71d5d935b.png
[/img]
[/center]
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that
$2a_n-1$ is a perfect square.
Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.
For geometric series $(a_n)$ with all items real, if $S_{10}=10,S_{30}=70$, then $S_{40}=$
$\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400$
Note: $S_n=\sum_{i=1}^{n}a_i$.
A set $ X$ of positive integers is called [i]nice[/i] if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a \plus{} b$ and $ |a \minus{} b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008.
[i]Author: Fedor Petrov[/i]
If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the
arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ ,
\[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]
In the fraction below and its decimal notation (with period of length $ 4$) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction:
$ \frac{ADA}{KOK}\equal{}0.SNELSNELSNELSNEL...$
$ Note.$ Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".
Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quantity of money. How much money does each brother have?
Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.
Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$.
Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.
If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.
Let $\mathbb{Z}^{+}$ be the set of positive integers.
[b]a)[/b] Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$, strictly increasing, such that
$f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$.
[b]b)[/b] For the function in [b]a[/b]. Prove that for every $n\in \mathbb{Z}^{+}$
$\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$
[b]c) [/b] Prove that in each inequality side of [b]b[/b] the equality can reach by infinite positive integers $n$.
Consider $n\geqslant 6$ coplanar lines, no two parallel and no three concurrent. These lines split the plane into unbounded polygonal regions and polygons with pairwise disjoint interiors. Two polygons are non-adjacent if they do not share a side. Show that there are at least $(n-2)(n-3)/12$ pairwise non-adjacent polygons with the same number of sides each.
Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.
The sequence $ (x_n)_{n \geq 1}$ is defined by:
$ x_1\equal{}1$
$ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$
Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.