Found problems: 85335
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$
[i]Proposed by Milan Haiman[/i]
For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?
Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.
Decide whether $S_n$ or $T_n$ is larger, where
\[S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}\]
Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$, $1\leq k\leq 10$, for which this is possible.
If in applying the quadratic formula to a quadratic equation
\[ f(x)\equiv ax^2 \plus{} bx \plus{} c \equal{} 0,
\]
it happens that $ c \equal{} \frac {b^2}{4a}$, then the graph of $ y \equal{} f(x)$ will certainly:
$ \textbf{(A)}\ \text{have a maximum} \qquad\textbf{(B)}\ \text{have a minimum} \qquad\textbf{(C)}\ \text{be tangent to the x \minus{} axis} \\
\qquad\textbf{(D)}\ \text{be tangent to the y \minus{} axis} \qquad\textbf{(E)}\ \text{lie in one quadrant only}$
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
Positive reals $a,b,c \leq 1$ satisfy $\frac{a+b+c-abc}{1-ab-bc-ca} = 1$. Find the minimum value of $$\bigg(\frac{a+b}{1-ab} + \frac{b+c}{1-bc} + \frac{c+a}{1-ca}\bigg)^{2}$$
Proposed by Harry Chen (Extile)
The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ .
Find $a$ and $b $ .
Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.
Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities
$$a + b + c> 0,$$
$$ ab + bc + ca > 0$$
$$ abc > 0$$
then $a > 0, b > 0, c > 0$.
Construct a right triangle given the hypotenuse and the median drawn to the leg.
For every \(i \in \mathbb{N}\) let \(A_i\), \(B_i\) and \(C_i\) be three finite and pairwise disjoint subsets of \(\mathbb{N}\). Suppose that for every pairwise disjoint sets \(A\), \(B\) and \( C\) with union \(\mathbb N\) there exists \(i\in \mathbb{N}\) such that \(A_i \subset A\), \(B_i \subset B\) and \(C_i \subset C\). Prove that there also exists a finite \(S\subset \mathbb{N}\) such that for every pairwise disjoint sets \(A\), \(B\) and \(C\) with union $\mathbb N$ there exists \(i\in S\) such that \(A_i \subset A\), \(B_i \subset B\) and \(C_i \subset C\).
[i]Submitted by András Imolay, Budapest[/i]
On a blackboard there are $2010$ natural nonzero numbers. We define a "move" by erasing $x$ and $y$ with $y\neq0$ and replacing them with $2x+1$ and $y-1$, or we can choose to replace them by $2x+1$ and $\frac{y-1}{4}$ if $y-1$ is divisible by 4.
Knowing that in the beginning the numbers $2006$ and $2008$ have been erased, show that the original set of numbers cannot be attained again by any sequence of moves.
Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?
Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and
$$ e_n=a + \prod_{k=0}^{n-1} e_k$$
for $n \geq 1$.
Prove that
(a) There exist infinitely many prime numbers that divide one element of the sequence.
(b) There exists one prime number that does not divide an element of the sequence.
(Theresia Eisenkölbl)
If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.
Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.
Find $A$ so that the ratio of $3\frac23$ to $22$ is the same as the ratio of $7\frac56$ to $A$
We are placing rooks on a $n \cdot n$ chess table that providing this condition:
Every two rooks will threaten an empty square at least.
What is the most number of rooks?
Let $n$ be a given positive integer. The sequence of real numbers $a_1, a_2, a_3, \cdots, a_n$ satisfy for each $m \leq n$, $$\left|\sum_{k=1}^m\frac{a_k}k\right| \leq 1.$$ Given this information, find the greatest possible value of $\left|\sum_{k=1}^n a_k\right|$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 13)[/i]
How many sigma $(\sigma)$ and pi $(\pi)$ bonds are in a molecule of ethyne (acetylene), $\ce{HCCH}?$
$ \textbf{(A) } 1 \sigma \text{ and } 1 \pi \qquad\textbf{(B) }2 \sigma \text{ and } 1 \pi \qquad\textbf{(C) }2 \sigma \text{ and } 3\pi \qquad\textbf{(D) }3 \sigma \text{ and } 2 \pi\qquad$