Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

Determine all triples $(x, y, z)$ of non-zero numbers such that \[ xy(x + y) = yz(y + z) = zx(z + x). \]

Mary is about to pay for five items at the grocery store. The prices of the items are $ \$$7.99, $ \$$ 4.99, $ \$$2.99, $ \$$1.99, and $ \$$0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $ \$$20.00 that she will receive in change? $ \textbf{(A) } 5 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25$

For how many integer $ n$, $ P \equal{} n^4 \plus{} 4n^3 \plus{} 3n^2 \minus{} 2n \plus{} 7$ is prime? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases: (i) $10 \leq n \leq 99$ (ii) $100 \leq n \leq 999$ (iii) $1000 \leq n \leq 9999$ (iv) $10000 \leq n \leq 99999$

Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ? The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.

The area of polygon $ ABCDEF$, in square units, is [asy]draw((0,0)--(4,0)--(4,9)--(-2,9)--(-2,4)--(0,4)--cycle); label("A",(-2,9),NW); label("B",(4,9),NE); label("C",(4,0),SE); label("D",(0,0),SW); label("E",(0,4),NE); label("F",(-2,4),SW); label("5",(-2,6.5),W); label("4",(2,0),S); label("9",(4,4.5),E); label("6",(1,9),N); label("All angles in this diagram are right.",(0,-3),S);[/asy] \[ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74 \]

In the acute triangle $ABC$, the bisectors of $A,B$ and $C$ intersect the circumcircle again at $A_1,B_1$ and $C_1$, respectively. Let $M$ be the point of intersection of $AB$ and $B_1C_1$, and let $N$ be the point of intersection of $BC$ and $A_1B_1$. Prove that $MN$ passes through the incentre of $\triangle ABC$.

Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.

a.) There are $2n+1$ ($n>2$) batteries. We don't know which batteries are good and which are bad but we know that the number of good batteries is greater by $1$ than the number of bad batteries. A lamp uses two batteries, and it works only if both of them are good. What is the least number of attempts sufficient to make the lamp work? b.) The same problem but the total number of batteries is $2n$ ($n>2$) and the numbers of good and bad batteries are equal. [i]Proposed by Alexander Shapovalov[/i]

You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

Let $ R\equal{}R_1\oplus R_2$ be the direct sum of the rings $ R_1$ and $ R_2$, and let $ N_2$ be the annihilator ideal of $ R_2$ (in $ R_2$). Prove that $ R_1$ will be an ideal in every ring $ \widetilde{R}$ containing $ R$ as an ideal if and only if the only homomorphism from $ R_1$ to $ N_2$ is the zero homomorphism. [Gy. Hajos]

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.

Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.

Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$. [i]Proposed by Qu Zhenhua[/i]

Let $A_1A_2A_3A_4$, $B_1B_2B_3B_4$, and $C_1C_2C_3C_4$ be three regular tetrahedra in $3$-dimensional space, no two of which are congruent. Suppose that, for each $i\in \{1,2,3,4\}$, $C_i$ is the midpoint of the line segment $A_iB_i$. Determine whether the four lines $A_1B_1$, $A_2B_2$, $A_3B_3$, and $A_4B_4$ must concur.

Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.

Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent. [hide=MOP 97 problem]Let $ABC$ be a triangle, and $M$, $N$, $P$ the points where its incircle touches the sides $BC$, $CA$, $AB$, respectively. The parallel to $AB$ through $N$ meets $MP$ at $E$, and the parallel to $AB$ through $M$ meets $NP$ at $F$. Prove that the lines $CP$, $MN$, $EF$ are concurrent. [url=https://artofproblemsolving.com/community/c6h22324p143462]also[/url][/hide]

Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

The Purple Plant Garden Store sells grass seed in ten-pound bags and fifteen-pound bags. Yesterday half of the grass seed they had was in ten-pound bags. This morning the store received a shipment of 27 more ten-pound bags, and now they have twice as many ten-pound bags as fifteen-pound bags. Find the total weight in pounds of grass seed the store now has.

Let be $z_1$ and $z_2$ two complex numbers such that $|z_1+2z_2|=|2z_1+z_2|$.Prove that for all real numbers $a$ is true $|z_1+az_2|=|az_1+z_2|$