Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are constructed on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.) a) Prove that the circles circumscribed around the outer triangles intersect in one point. b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point

Let $n$ be a positive integer. The following $35$ multiplication are performed: $$1 \cdot n, 2 \cdot n, \dots, 35 \cdot n.$$ Show that in at least one of these results the digit $7$ appears at least once.

On side $AB$ of triangle $ABC$, points $M$ and $K$ are taken ($M$ on segment $AK$). It is known that $AM: MK: MB = a: b: c$. Straight lines $CM$ and $CK$ intersect for the second time the circumscribed circle of the triangle $ABC$ at points $E$ and $F$, respectively. In what ratio does the circumscribed circle of the triangle $BMF$ divide the segment $BE$?

$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$. $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$Find the maximum number of perfect squares from $S_i$

Let $p(x)=x^4-4x^3+2x^2+ax+b$. Suppose that for every root $\lambda$ of $p$, $\frac{1}{\lambda}$ is also a root of $p$. Then $a+b=$ [list=1] [*] -3 [*] -6 [*] -4 [*] -8 [/list]

Given points $M$ and $N$, the bases of heights $AM$ and $BN$ of $\vartriangle ABC$ and the line to which the side $AB$ belongs. Construct $\vartriangle ABC$.

Find all positive integers $n$ such that the number \[A=\sqrt{\frac{9n-1}{n+7}}\] is rational.

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

Compute the value of $1^{25}+2^{24}+3^{23}+\ldots+24^2+25^1$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor$.

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$

[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$? [b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$? [b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property? [b]p4.[/b] What is the next number in the following geometric series: $1020100$, $10303010$, $104060401$? [b]p5.[/b] A circle has radius $A$ and area $r$. If $A = r^2\pi$, then what is the diameter, $C$, of the circle? [b]p6.[/b] If $$O + N + E = 1$$ $$T + H + R + E + E = 3$$ $$N + I + N + E = 9$$ $$T + E + N = 10$$ $$T + H + I + R + T + E + E + N = 13$$ Then what is the value of $O$? [b]p7.[/b] By shifting the initial digit, which is $6$, of the positive integer $N$ to the end (for example, $65$ becomes $56$), we obtain a number equal to $\frac{N}{4}$ . What is the smallest such $N$? [b]p8.[/b] What is $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$ ? [b]p9.[/b] How many permutations of the digits of $1234$ are divisible by $11$? [b]p10.[/b] If you choose $4$ cards from a normal $52$ card deck (with replacement), what is the probability that you will get exactly one of each suit (there are $4$ suits)? [b]p11.[/b] If $LMT$ is an equilateral triangle, and $MATH$ is a square, such that point $A$ is in the triangle, then what is $HL/AL$? [b]p12.[/b] If $$\begin{tabular}{cccccccc} & & & & & L & H & S\\ + & & & & H & I & G & H \\ + & & S & C & H & O & O & L \\ \hline = & & S & O & C & O & O & L \\ \end{tabular}$$ and $\{M, A, T,H, S, L,O, G, I,C\} = \{0, 1, 2, 3,4, 5, 6, 7, 8, 9\} $, then what is the ordered pair $(M + A +T + H, [T + e + A +M])$ where $e$ is $2.718...$and $[n]$ is the greatest integer less than or equal to $n$ ? [b]p13.[/b] There are $5$ marbles in a bag. One is red, one is blue, one is green, one is yellow, and the last is white. There are $4$ people who take turns reaching into the bag and drawing out a marble without replacement. If the marble they draw out is green, they get to draw another marble out of the bag. What is the probability that the $3$rd person to draw a marble gets the white marble? [b]p14.[/b] Let a "palindromic product" be a product of numbers which is written the same when written back to front, including the multiplication signs. For example, $234 * 545 * 432$, $2 * 2 *2 *2$, and $14 * 41$ are palindromic products whereas $2 *14 * 4 * 12$, $567 * 567$, and $2* 2 * 3* 3 *2$ are not. 2015 can be written as a "palindromic product" in two ways, namely $13 * 5 * 31$ and $31 * 5 * 13$. How many ways can you write $2016$ as a palindromic product without using 1 as a factor? [b]p15.[/b] Let a sequence be defined as $S_n = S_{n-1} + 2S_{n-2}$, and $S_1 = 3$ and $S_2 = 4$. What is $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$ ? [b]p16.[/b] Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$. [b]p17.[/b] Evaluate $\dfrac{8+ \dfrac{8+ \dfrac{8+...}{3+...}}{3+ \dfrac{8+...}{3+...}}}{3+\dfrac{8+ \dfrac{8+...}{3+...}}{ 3+ \dfrac{8+...}{3+...}}}$, assuming that it is a positive real number. [b]p18.[/b] $4$ non-overlapping triangles, each of area $A$, are placed in a unit circle. What is the maximum value of $A$? [b]p19.[/b] What is the sum of the reciprocals of all the (positive integer) factors of $120$ (including $1$ and $120$ itself). [b]p20.[/b] How many ways can you choose $3$ distinct elements of $\{1, 2, 3,...,4000\}$ to make an increasing arithmetic series? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, $E$ and $F$ the points where the incircle and $A$-excircle touch $AB$, and $D$ the point on $BC$ such that the triangles $ABD$ and $ACD$ have equal in-radii. The lines $DB$ and $DE$ intersect the circumcircle of triangle $ADF$ again in the points $X$ and $Y$. Prove that $XY\parallel AB$ if and only if $AB=AC$.

The sides of a triangle are in the ratio $ 6: 8: 9$. Then: $ \textbf{(A)}\ \text{the triangle is obtuse} \qquad\textbf{(B)}\ \text{the angles are in the ratio } 6: 8: 9$ $ \textbf{(C)}\ \text{the triangle is acute}$ $ \textbf{(D)}\ \text{the angle opposite the largest side is double the angle opposite the smallest side}$ $ \textbf{(E)}\ \text{none of these}$

Let $a$ and $b$ be two integers which are coprime and let $n$ be one variable integer. Determine probability that number of solutions $(x,y)$, where $x$ and $y$ are nonnegative integers, of equation $ax+by=n$ is $\left\lfloor \frac{n}{ab} \right\rfloor + 1$

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

\[\int_0^1 \frac{(x+1)\log(x)}{x^3-1}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.

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 define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that : $i)$$f(p)=1$ for all prime numbers $p$. $ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$ find the smallest $n \geq 2016$ such that $f(n)=n$

Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that $$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$

Let $ABC$ be an isosceles triangle with $AC = BC$. On the arc $CA$ of its circumcircle, which does not contain $ B$, there is a point $ P$. The projection of $C$ on the line $AP$ is denoted by $E$, the projection of $C$ on the line $BP$ is denoted by $F$. Prove that the lines $AE$ and $BF$ have equal lengths. (W. Janous, WRG Ursulincn, Innsbruck)

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

Prove that in each polyhedron there exist two faces with the same number of edges.

In triangle $ABC$ with $AB = 7$, $AC = 8$, and $BC = 9$, the $A$-excircle is tangent to $BC$ at point $D$ and also tangent to lines $AB$ and $AC$ at points $ $ and $F$, respectively. Find $[DEF]$. (The $A$-excircle is the circle tangent to segment $BC$ and the extensions of rays $AB$ and $AC$. Also, $[XY Z]$ denotes the area of triangle $XY Z$.)

Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors.