Found problems: 85335
Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions:
a)$ \mathbb{Z} \varsubsetneqq I$;
b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$);
c) $ I$ only has trivial endomorphisms.
[i]E. Fried[/i]
Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$.
[i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]
Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a move, he loses the game. The game is played once on each $6\times 7$, $6 \times 8$, $7 \times 7$, $7 \times 8$, and $8 \times 8$ chessboard. In how many of them, can the first player guarantee to win?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None}
$
How many positive integers less than $1$ million have exactly $5$ positive divisors?
$$
\mathrm a. ~ 1\qquad \mathrm b.~5\qquad \mathrm c. ~11 \qquad \mathrm d. ~23 \qquad \mathrm e. ~24
$$
Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$.
Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$
[i]Radu Bumbăcea & Călin Popescu[/i]
Let $E$, $F$, $G$ be points on sides $[AB]$, $[BC]$, $[CD]$ of the rectangle $ABCD$, respectively, such that $|BF|=|FQ|$, $m(\widehat{FGE})=90^\circ$, $|BC|=4\sqrt 3 / 5$, and $|EF|=\sqrt 5$. What is $|BF|$?
$
\textbf{(A)}\ \dfrac{\sqrt{10} - \sqrt{2}}{2}
\qquad\textbf{(B)}\ \sqrt 3 -1
\qquad\textbf{(C)}\ \sqrt 3
\qquad\textbf{(D)}\ \dfrac{\sqrt{11} - \sqrt{3}}{2}
\qquad\textbf{(E)}\ 1
$
Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.
A $5779$-dimensional polytope is call a [b]$k$-tope[/b] if it has exactly $k$ $5778$-dimensional faces.
Find all sequences $b_{5780}, b_{5781}, \dots, b_{11558}$ of nonnegative integers, not all $0$, such that the following condition holds:
It is possible to tesselate every $5779$-dimensional polytope with [u]convex[/u] $5779$-dimensional polytopes, such that the number of $k$-topes in the tessellation is proportional to $b_k$, while there are no $k$-topes in the tessellation if $k\notin \{5780, 5781, \dots, 11558\}$.
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
Determine all the integers $a$ and $b$, such that $\sqrt{2010 + 2 \sqrt{2009}}$ be a solution of the equation $x^2 + ax + b = 0$. Prove that for such $a$ and $b$ the number$\sqrt{2010 - 2 \sqrt{2009}}$ is not a solution to the given equation.
In two wooden boxes, there are $1994$ and $2024$ marbles, respectively. Spiro and Cvetko play the following game: alternately, each player takes a turn and removes some marbles from one of the boxes, so that the number of removed marbles in that turn is a divisor of the current number of marbles in the other box. The winner of the game is the one after whose turn both boxes are empty. Spiro takes the first turn. Which of the players has a winning strategy?
For every nonempty set of real numbers $S{}$ denote $\sigma(S)$ the sum of its elements. Let $A{}$ be a set of $n{}$ positive integers. Show that the set of all sums $\sigma{}$ of all nonempty sets of $A{}$ can be partitioned in $n{}$ groups such that the ratio between the greatest number and the smallest number from each group is less than $2$.
The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that:
a) $QR = AD$,
b) $QR \perp AD$.
Prove that the following inequality holds for each natural number $ n$.
\[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that
\[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]
Compute the value of the expression
\[ 2009^4 \minus{} 4 \times 2007^4 \plus{} 6 \times 2005^4 \minus{} 4 \times 2003^4 \plus{} 2001^4 \, .\]
How many possible arrangements of bishops are there on a $8 \times 8$ chessboard such that no bishop threatens a square on which another lies and the maximum number of bishops are used? (Note that a bishop threatens any square along a diagonal containing its square.)
Andrew starts with the $2018$-tuple of binary digits $(0,0,\dots,0)$. On each turn, he randomly chooses one index (between $1$ and $2018$) and flips the digit at that index (makes it $1$ if it was a $0$ and vice versa). What is the smallest $k$ such that, after $k$ steps, the expected number of ones in the sequence is greater than $1008?$
You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$
Problem 6. Let H be orthocenter of an acute-angled triangle ABC. Points E,F are on the segments AB,AC respectively, such that BE=BH,CF=CH. The lines EH,FH meet BC in X,Y respectively. Draw the perpendicular HZ from H to EF. Prove that the circumcircle of triangle XYZ is tangent to the circle with diameter BC.
A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called [i]clocky[/i] if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.
Points $A,B,C$ lie on a circle $\omega$ such that $BC$ is a diameter. $AB$ is extended past $B$ to point $B'$ and $AC$ is extended past $C$ to point $C'$ such that line $B'C'$ is parallel to $BC$ and tangent to $\omega$ at point $D$. If $B'D = 4$ and $C'D = 6$, compute $BC$.