In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a [i]blue triangle[/i]. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is [i]attached[/i] to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?

Kylie is trying to count to $202250$. However, this would take way too long, so she decides to only write down positive integers from $1$ to $202250$, inclusive, that are divisible by $125$. How many times does she write down the digit $2$?

The four faces of a right triangular pyramid are equilateral triangles whose edge measures $3$ dm. Suppose the pyramid is hollow, resting on one of its faces at a horizontal surface (see attached figure) and that there is $2$ dm$^3$ of water inside. Determine the height that the liquid reaches inside the pyramid. [img]https://cdn.artofproblemsolving.com/attachments/0/7/6cd6e1077620371e56ed57d19fd3d05a58904e.png[/img]

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$. (E. Diomidov)

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.

Edges of a complete graph with $2m$ vertices are properly colored with $2m-1$ colors. It turned out that for any two colors all the edges colored in one of these two colors can be described as union of several $4$-cycles. Prove that $m$ is a power of $2$.

Let $A_1A_2A_3A_4A_5$ be a regular pentagon inscribed in a circle with area $\tfrac{5+\sqrt{5}}{10}\pi$. For each $i=1,2,\dots,5$, points $B_i$ and $C_i$ lie on ray $\overrightarrow{A_iA_{i+1}}$ such that \[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices are taken modulo 5. The value of $\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}$ (where $[\mathcal P]$ denotes the area of polygon $\mathcal P$) can be expressed as $\tfrac{a+b\sqrt{5}}{c}$, where $a$, $b$, and $c$ are integers, and $c > 0$ is as small as possible. Find $100a+10b+c$. [i]Proposed by Robin Park[/i]

For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds \[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \] [i]Li Shenghong[/i]

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?

[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set $ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $ and construct the following matrices of order $n$: $ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $ where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$: $ S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix} $ ($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$). Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]

We will say that a natural number is [i]acceptable[/i] if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A [i]legal move[/i] consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

A group of men, some of them accompanied by their wives, spent $\$1.000$ on a hotel. Each man spent $\$19$ and each woman $\$13$. Determine how many women and how many men there were.

Some figures stand in certain cells of a chess board. It is known that a figure stands on each row, and that different rows have a different number of figures. Prove that it is possible to mark $8$ figures so that on each row and column stands exactly one marked figure.

Find the value of $a$ such that : \[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers?

Find the largest natural number $n$ such that \[2^n + 2^{11} + 2^8\] is a perfect square.

Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.

Known $f:\mathbb{N}_0 \to \mathbb{N}_0$ function for $\forall x,y\in \mathbb{N}_0$ the following terms are paid $(a). f(0,y)=y+1$ $(b). f(x+1,0)=f(x,1)$ $(c). f(x+1,y+1)=f(x,f(x+1,y)).$ Find the value if $f(4,1981)$

At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? $\text{(A)}\ \dfrac{1}{16} \qquad \text{(B)}\ \dfrac{3}{20} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{17}{20} \qquad \text{(E)}\ \dfrac{9}{10}$

Find the least positive integers $m,k$ such that a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube. b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.