Found problems: 85335
If \[x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2},\] where $ i^2 \equal{} \minus{} 1$, then which of the following is [i]not[/i] correct?
$ \textbf{(A)}\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad \textbf{(B)}\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad \textbf{(C)}\ x^9 \plus{} y^9 \equal{} \minus{} 1$
$ \textbf{(D)}\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad \textbf{(E)}\ x^{13} \plus{} y^{13} \equal{} \minus{} 1$
If $x_0=x_1=1$, and for $n\geq1$
$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,
find a formula for $x_n$ as a function of $n$.
Emily has two cups $A$ and $B$, each of which can hold $400$ mL, A initially with $200$ mL of water and $B$ initially with $300$ mL of water. During a round, she chooses the cup with more water (randomly picking if they have the same amount), drinks half of the water in the chosen cup, then pours the remaining half into the other cup and refills the chosen cup to back to half full. If Emily goes for $20$ rounds, how much water does she drink, to the nearest integer?
Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10}
, \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$
Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds:
$$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$
Let $n,s,t$ be three positive integers, and let $A_1,\ldots, A_s, B_1,\ldots, B_t$ be non-necessarily distinct subsets of $\{1,2,\ldots,n\}$. For any subset $S$ of $\{1,\ldots,n\}$, define $f(S)$ to be the number of $i\in\{1,\ldots,s\}$ with $S\subseteq A_i$ and $g(S)$ to be the number of $j\in\{1,\ldots,t\}$ with $S\subseteq B_j$. Assume that for any $1\leq x<y\leq n$, we have $f(\{x,y\})=g(\{x,y\})$. Show that if $t<n$, then there exists some $1\leq x\leq n$ so that $f(\{x\})\geq g(\{x\})$.
[i]Proposed by usjl[/i]
Prova that any integer $ Z $ has a unique representation
$$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$
where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $
Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is
$\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$
The marked price of a book was $ 30\%$ less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?
$ \textbf{(A)}\ 25\% \qquad
\textbf{(B)}\ 30\% \qquad
\textbf{(C)}\ 35\% \qquad
\textbf{(D)}\ 60\% \qquad
\textbf{(E)}\ 65\%$
Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$.
Determine the minimum value of the red number at the end of this procedure.
Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers. Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$ such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \]
Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that $a_k \in A$ for all $k \ge 2025$ and, for every positive integer $d$, there exist infinitely many positive integers $n$ satisfying
\[ \sum_{k=1}^n a_k k^d = 0. \]
[i]Daniel Zhu[/i]
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)
[i](Karl Czakler)[/i]
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
Given three distinct digits a, b and c, it is possible, by choosing two digits at a time, to form six two-digit numbers. Determine all possible sets {a, b, c} for which the sum of the six two-digits numbers is 484.
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$
Determine if there exist non-constant polynomials $P(x)$, $Q(x)$ and $R(x)$ with real coefficients and leading coefficient $1$, such that each of the polynomials
\[
P(Q(x)), \quad Q(R(x)), \quad R(P(x))
\]
has at least one real root, while each of the polynomials
\[
Q(P(x)), \quad R(Q(x)), \quad P(R(x))
\]
has no real roots.
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
Solve the equation: $\sin ^3x+\sin ^32x+\sin ^33x=(\sin x + \sin 2x + \sin 3x)^3$.
A circle of radius $ r$ is inscribed in a right isosceles triangle, and a circle of radius $ R$ is circumscribed about the triangle. Then $ R/r$ equals
$ \textbf{(A)}\ 1\plus{}\sqrt2\qquad
\textbf{(B)}\ \frac{2\plus{}\sqrt2}2 \qquad
\textbf{(C)}\ \frac{\sqrt2\minus{}1}2 \qquad$
$ \textbf{(D)}\ \frac{1\plus{}\sqrt2}2 \qquad
\textbf{(E)}\ 2(2\minus{}\sqrt2)$
Five points $O,A,B,C,D$ are taken in order on a straight line with distances $OA=a$, $OB=b$, $OC=c$, and $OD=d$. $P$ is a point on the line between $B$ and $C$ and such that $AP:PD=BP:PC$. Then $OP$ equals:
$\text{(A)}\ \dfrac{b^2-bc}{a-b+c-d} \qquad
\text{(B)}\ \dfrac{ac-b}{a-b+c-d} \qquad
\text{(C)}\ -\dfrac{bd+c}{a-b+c-d}\qquad\\
\text{(D)}\ \dfrac{bc+ad}{a+b+c+d}\qquad
\text{(E)}\ \dfrac{ac-bd}{a+b+c+d} \qquad$
Let $a,b,c$ be three positive real numbers such that $a+ b +c =1$.
Let $\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}$
Prove that the roots of $x^2 + x + 4 \lambda = 0$ are real.
Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae:
$$S=\frac k2+\ell-1$$
[i]V. Chukanov[/i]
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$.
Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.
Show that any subset of $ V\equal{} \{ 1,2,...,24,25 \}$ with $ 17$ or more elements contains at least two distinct numbers the product of which is a perfect square.