Let $a,b,c$ be distinct positive integers and let $r,s,t$ be positive integers such that: $ab+1=r^2,ac+1=s^2,bc+1=t^2$ Prove that it is not possible that all three fractions$ \frac{rt}{s}, \frac{rs}{t}, \frac{st}{r}$ are integers.

Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?

Does there exist a positive integer $n>10^{100}$, such that $n^2$ and $(n+1)^2$ satisfy the following property: every digit occurs equal number of times in the decimal representations of each number?

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? $ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$

A board has $2$, $4$, and $6$ written on it. A person repeatedly selects (not necessarily distinct) values for $x$, $y$, and $z$ from the board, and writes down $xyz+xy+yz+zx+x+y+z$ if and only if that number is not yet on the board and is also less than or equal to $2013$. This person repeats this process until no more numbers can be written. How many numbers will be written at the end of the process?

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's [i]vicious hobgoblin +1 of viciousness,[/i] Hugo rolls $25$ $20$-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers was as large as possible. How many $2$s did Hugo roll?

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Find all the positive integers $p$ with the property that the sum of the first $p$ positive integers is a four-digit positive integer whose decomposition into prime factors is of the form $2^m3^n(m + n)$, where $m, n \in N^*$.

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

Find all functions $f:\mathbb R\to\mathbb R$ such that $$(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)$$for all $x, y\in\mathbb R$.

Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles? $\text{(A) }\frac{1}{16}\qquad\text{(B) }\frac{1}{9}\qquad\text{(C) }\frac{1}{8}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{4}$

Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$). (translated by L. Erdős)

For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following. $$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$ ($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)

Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]

Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Determine the largest possible constant \( C \) such that for any positive real numbers \( x, y, z \), which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] [i]Proposed by Vadym Solomka[/i]

A circle $\mathcal C$ and five distinct points $M_1,M_2,M_3,M_4$ and $M$ on $\mathcal C$ are given in the plane. Prove that the product of the distances from $M$ to lines $M_1M_2$ and $M_3M_4$ is equal to the product of the distances from $M$ to the lines $M_1M_3$ and $M_2M_4$. What can one deduce for $2n+1$ distinct points $M_1,\ldots,M_{2n},M$ on $\mathcal C$?

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.