Found problems: 85335
Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?
Is the following statement true?
For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.
A four-digit number $YEAR$ is called [i]very good[/i] if the system
\begin{align*}
Yx+Ey+Az+Rw& =Y\\
Rx+Yy+Ez+Aw & = E\\\
Ax+Ry+Yz+Ew & = A\\
Ex+Ay+Rz+Yw &= R
\end{align*}
of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century.
(The $21$st century starts in $2001$ and ends in $2100$.)
[i]Proposed by Tomáš Bárta, Charles University, Prague[/i]
Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$.
A die is an unitary cube with numbers from $1$ to $6$ written on its faces, so that each number appears once and the sum of the numbers on any two opposite faces is $7$. We construct a large $3 \cdot 3 \cdot 3$ cube using$ 27$ dice. Find all possible values of the sum of numbers which can be seen on the faces of the large cube.
Let $ n$ be a positive integer such that $ \tfrac{1}{2}\plus{}\tfrac{1}{3}\plus{}\tfrac{1}{7}\plus{}\tfrac{1}{n}$ is an integer. Which of the following statements is [b]not[/b] true?
$ \textbf{(A)}\ 2\text{ divides }n \qquad
\textbf{(B)}\ 3\text{ divides }n \qquad
\textbf{(C)}\ 6\text{ divides }n \qquad
\textbf{(D)}\ 7\text{ divides }n \\
\textbf{(E)}\ n>84$
Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$.
Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point
on $\Gamma$ such that $IN \perp MT$. Finally, let $P $ and $Q$ be the intersection points of $TB $ and $TC$,
respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$.
[i]Proposed by Patrik Bak - Slovakia[/i]
$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$
prove that $x$ is a root of a quadratic equation with integer coefficients.
What is the hybridization of the carbon atom in a carboxyl group?
$ \textbf{(A) }sp \qquad\textbf{(B) } sp^2\qquad\textbf{(C) } sp^3\qquad\textbf{(D) } dsp^3\qquad $
9.2 Given 19 cards. Is it possible to write a nonzero digit on each card in such a way that you can compose from these cards an unique 19-digits number, which is divisible by 11?
([i]R. Zhenodarov, I. Bogdanov[/i])
Let $a>0$ and $\mathcal{F} = \{f:[0,1] \to \mathbb{R} : f \text{ is concave and } f(0)=1 \}.$ Determine $$\min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}.$$
An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$.
Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.
Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$.
Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$ and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$.
If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear.
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.
Determine all natural numbers $n \ge 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers.
[i](Walther Janous)[/i]
How many (nondegenerate) tetrahedrons can be formed from the vertices of an $n$-dimensional hypercube?
Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.
Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.
In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$. Let $d>3$ be the biggest degree in the graph $G$. Prove that if :
$$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$
Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected.
Proposed by [i]Ali Mirzaie[/i]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x+f(y))-f(x)=(x+f(y))^4-x^4\]
for all $x,y \in \mathbb{R}$.
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$.
[i]Proposed by Nikola Velov[/i]
Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.
Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.
What is the sum of all two-digit odd numbers whose digits are all greater than $6$?