Found problems: 85335
Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$
Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$
[i](Walther Janous)[/i]
Suppose $a_1,a_2,\cdots,a_8$ are eight distinct integers from $\{1,2,\cdots,16,17\}$. Show that there is an integer $k > 0$ such that the equation $a_i - a_j = k$ has at least three different solutions.
Also, find a specific set of 7 distinct integers from $\{1,2,\ldots,16,17\}$ such that the equation $a_i - a_j = k$ does not have three distinct solutions for any $k > 0$.
Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.
Let $ABCD$ be an isosceles trapezoid such that $AB=10$, $BC=15$, $CD=28$, and $DA=15$. There is a point $E$ such that $\triangle AED$ and $\triangle AEB$ have the same area and such that $EC$ is minimal. Find $EC$.
Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$
Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin:
$$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$
Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?
Prove it is possible to find $2^{2021}$ different pairs of positive integers $(a_i,b_i)$ such that:
$$ \frac{1}{a_ib_i}+\frac{1}{a_2b_2} + \ldots + \frac{1}{a_{2^{2021}}b_{2^{2021}}} = 1 $$
$$ a_1+a_2 +\ldots a_{2^{2021}} +b_1+b_2 + \ldots +b_{2^{2021}} = 3^{2022} $$
[b]Note: [/b]Pairs $(a,b)$ and $(c,d)$ are different if $a \neq c$ or $b \neq d$
Find all natural values of $k$ for which the system
$$\begin{cases}x_1+x_2+\ldots+x_k=9\\\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$
has solutions in positive numbers. Find these solutions.
[i]I. Dimovski[/i]
A child is sliding out of control with velocity $v_\text{c}$ across a frozen lake. He runs head-on into another child, initially at rest, with $3$ times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision?
(A) $2v_\text{c}$
(B) $v_\text{c}$
(C) $v_\text{c}/2$
(D) $v_\text{c}/3$
(E) $v_\text{c}/4$
Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$.
(A.Akopjan)
Let $x,y,z,t$ be positive numbers.Prove that
$\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$
Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?
The positive divisors of a positive integer $n$ are written in increasing order starting with 1.
\[1=d_1<d_2<d_3<\cdots<n\]
Find $n$ if it is known that:
[b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$
[b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$
Find all triplets natural numbers $(a, b, c)$ satisfied
\[GCD(a, b) + LCM(a,b) = 2021^c\]
with $|a - b|$ and $(a+b)^2 + 4$ are both prime number
What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
$\textbf{(A) } \text{It is a negative number ending with a 1.}$
$\textbf{(B) } \text{It is a positive number ending with a 1.}$
$\textbf{(C) } \text{It is a negative number ending with a 5.}$
$\textbf{(D) } \text{It is a positive number ending with a 5.}$
$\textbf{(E) } \text{It is a negative number ending with a 0.}$
For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.
Let the boxes in picture $1$ be marked as in picture $2$ below (from top to bottom in layers). In one move it is allowed to switch the empty box with another box adjacent to it (two boxes are adjacent if they share a common side). Can the arrangement of the numbers in picture $3$ be obtained after finitely many moves?
Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.
It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$.
a) Find the greatest value of $a$, such that this equation has at least one real root.
b) Find all the values of $a$, such that the equation has at least one real root.
Given $12$ segments, it is known that they can be divided into $4$ groups of $3$ segments each in such a way that a triangle can be formed from the segments of each triplet. Is it always possible to divide these $12$ segments into $3$ groups of $4$ segments each in such a way that a quadrilateral can be formed from the segments of each quartet?
[i]Proposed by Mykhailo Shtandenko[/i]
On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$, then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$. In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$.
The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.
In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$
[img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]
$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$. Prove that $AB < AC$.
A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?
Can every polynomial with integer coefficients be expressed as a sum of cubes
of polynomials with integer coefficients?
[hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]