Found problems: 85335
Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$. The minimal possible area of triangle $ABC$ can be expressed as $\frac{\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2$ dividing $m$. Compute $100m+n$.
[i]Proposed by Yannick Yao[/i]
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line$ AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD =16$, find the value of the product $r_1r_2$.
A triangle $ABC$ is inscribed in a circle. Let $A_1$ be the point diametrically opposed to $A$, $A_0$ be the midpoint of the side $BC$ and $A_2$ be the point symmetric to $A_1$ with respect to $A_0$; the points $B_2$ and $C_2$ are defined in a similar way starting from $B$ and $C$. Prove that the three points $A_2$, $B_2$ and $C_2$ coincide.
(A Jagubjanz)
The circle is divided into four non-overlapping gaps $ AB $, $ BC $, $ CD $ and $ DA $. Prove that the segment joining the midpoints of the arcs $AB$ and $CD$ is perpendicular to the segment joining the midpoints of the arcs $BC$ and $DA$.
Given $ a, b, c > 0$. Prove that: $ (1\plus{}a\plus{}b\plus{}c)(1\plus{}ab\plus{}bc\plus{}ca) \ge 4\sqrt{2}\sqrt{(a\plus{}bc)(b\plus{}ca)(c\plus{}ab)}$
:)
Find the total number of integer solutions of the equation $$x^5-y^2=4$$
[i](Erken)[/i]
Sally is going shopping for stuffed tigers. She finds $5$ orange, $10$ white, and $2$ cinnamon colored tigers. Sally decides to buy two tigers of different colors. Assuming all the tigers are distinct, in how many ways can she choose two tigers?
$a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)$
[b](a)[/b] Find $a_n$.
[b](b)[/b] Let $b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}$.
Find $S_n=\sum_{i=1}^nb_i$.
Let $a$ and $b$ be the roots of the equation $x^2-47x+289=0$. Compute $\sqrt{a}+\sqrt{b}$.
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation
\[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\]
Show that at least two of these integers are equal to each other.
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property:
if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.
Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
On Babba Island they use a two-letter alphabet, $a$ and $b$, and every (finite) sequence of letters is a word. For each set $P$ of six words of $4$ letters each, we denote $N_P$ to the set of all words that do not contain any of the words of $P$ as a syllable (subword).
Prove that if $N_P$ is finite, then all its words are of length less than or equal to $10$, and find a set $P$ such that $N_P$ is finite and contains at least one word of length $10$.
The entries of a $7 \times 5$ table are fi lled with numbers so that in each $2 \times 3$ rectangle (vertical or horizontal) the sum of numbers is $0$. For $100$ dollars Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend in order to learn the total sum of numbers in the table for sure?
If the difference between the greatest and the smallest root of the equation $x^3 - 6x^2 + 5$ is equal to $F$, which of the following is true?
$
\textbf{(A)}\ 0 \leq F < 2
\quad\textbf{(B)}\ 2 \leq F < 4
\quad\textbf{(C)}\ 4 \leq F < 6
\quad\textbf{(D)}\ 6 \leq F < 8
\quad\textbf{(E)}\ 8 \leq F
$
Prove that exists integer $n > 2003$ that in sequence
$\binom{n}{0}$, $\binom{n}{1}$, $\binom{n}{2}$, ..., $\binom{n}{2003}$
each element is a divisor of all elements which are after him.
How many ordered pairs of integers (x; y) satisfy the equation $x^2 + y^2 + xy = 4(x + y)?$.
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
[i]Victor Wang[/i]
Consider the square $ABCD$ and the point $E$ on the side $AB$. The line $DE$ intersects the line $BC$ at point $F$, and the line $CE$ intersects the line $AF$ at point $G$. Prove that the lines $BG$ and $DF$ are perpendicular.
a) On a circumference $8$ points are marked. We say that Juliana does an “T-operation ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do?
b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?
Let $ABC$ be acute triangle. Inscribe a rectangle $DEFG$ in this triangle such that $D\in AB,E\in AC,F\in BC,G\in BC$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$.
You want to sort the numbers 5 4 3 2 1 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, [i]6 5 3[/i] 4 2 1 -> 4 2 [i]6 5 3[/i] 1 is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 1 2 3 4 5?
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$
(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$
(Albania)
Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions:
(i) $6p$ does not divide $q-1$;
(ii) $q$ divides $x^2+xy+y^2$;
(iii) $q$ does not divide $x+y-z$.
Prove that $x^p+y^p\ne z^p$.