Found problems: 85335
Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks?
[asy]
unitsize(1cm);
draw(scale(3)*polygon(6));
filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray);
[/asy]
$\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}$
The triangle $ ABC$ has vertices in such manner that $ AB \equal{} 3, BC \equal{} 4,$ and $ AC \equal{} 5$. The inscribed circle is tangent to $ AB$ in $ C'$, $ BC$ in $ A'$ and $ AC$ in $ B'.$ What is the ratio between the area of the triangles $ A'B'C'$ and $ ABC$?
A. 1/4
B. 1/5
C. 2/9
D. 4/21
E. 5/24
Find all values of $ r$ such that the inequality $$r (ab + bc + ca) + (3- r) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \ge 9$$
is true for $a,b,c$ arbitrary positive reals
A one-person game with two possible outcomes is played as follows:
After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$
Prove that for any distinct rational numbers $a, b, c$, the number \[\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}+\frac{1}{(a-b)^{2}}\] is the square of some rational number.
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$
In which cases do we have equality?
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube.
[i]Dinu Serbanescu[/i]
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$
where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system.
Show that there is an integer $k$, with no $9$ in it's decimal representation, such that:
$$S(2^{24^{2017}}k)=S(k)$$
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
An equilateral triangle $ABC$ has side length $7$. Point $P$ is in the interior of triangle $ABC$, such that $PB=3$ and $PC=5$. The distance between the circumcenters of $ABC$ and $PBC$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime and $m$ and $p$ are relatively prime positive integers. What is $m+n+p$?
A graph consists of a set of vertices, some of which are connected by (undirected) edges. A [i]star[/i] of a graph is a set of edges with a common endpoint. A [i]matching[/i] of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph $G$ is larger than $2(k-1)^2$, then $G$ contains a matching of size $k$ or a star of size $k$.
Each of the cells of a table 2014 x 2014 is colored in white or black. It is known that each square 2 x 2 contains an even number of black cells and each cross (3 x 3 square without its corner cells) contains an odd number of black cells. Prove that the 4 corner cells of the table are in the same color.
Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:
i) $g(2013)=g(2014) = 0,$
ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$
Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$
For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which
\[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Bob is practicing addition in base $2.$ Each time he adds two numbers in base $2,$ he counts the number of carries. For example, when summing the numbers $1001$ and $1011$ in base $2,$
\[\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\
0&1&0&0&1\\0&1&0&1&1 \\ \hline
1&0&1&0&0
\end{array}\]
there are three carries (shown on the top row). Suppose that Bob starts with the number $0,$ and adds $111~($i.e. $7$ in base $2)$ to it one hundred times to obtain the number $1010111100~($i.e. $700$ in base $2).$ How many carries occur (in total) in these one hundred calculations?
\[\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297\]
Let $M$ be the (tridiagonal) $10\times10$ matrix
$$M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}$$Show that $M$ has exactly nine positive real eigenvalues (counted with multiplicities).
A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normalor inversive. If USJL walks over an inversive door, he would become inverted-USJL,and vice versa. USJL must choose a room to begin and walk pass each room exactly once. If it is inverted-USJL showing up after finishing, then he would be trapped for all eternity. Prove that USJL could always escape.
Sabine has a very large collection of shells. She decides to give part of her collection to her sister.
On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day.
The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time.
What are the possible numbers of shells that Sabine could have had in the very beginning?
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?
Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$.
Prove that $(PS)$ and $(QR)$ are perpendicular.
Quadrilateral$ ABCD$ has two diagonals $AC,BD$ that are mutually perpendicular. Let $M$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$. Suppose that $L$ is the intersection of two circles $(MAC)$ and $(MBD)$. Prove that the circumcenters of triangles $LAB,LBC,LCD,LDA$ are on the same circle called $\omega$ and that three circles $(MAC), (MBD), \omega$ are pairwise orthogonal.
Nguyễn Văn Linh