Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy]import three; import solids; size(8cm); currentprojection=orthographic(-1,-5,3); picture lpic, rpic; size(lpic,5cm); draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight); draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight); draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight); draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8)); draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed); draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed); draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0)); draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8)); size(rpic,5cm); draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight); draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed); draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)); draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8)); draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8)); label(rpic,"$A$",(-3,3*sqrt(3),0),W); label(rpic,"$B$",(-3,-3*sqrt(3),0),W); add(lpic.fit(),(0,0)); add(rpic.fit(),(1,0));[/asy]

Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$ for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

Let $m,n,p$ be three positive integers, and let $m'=\gcd(m,np)$, $n'=\gcd(n,pm)$ and $p'=\gcd(p,mn)$. Prove that the equation $x^m+y^n=z^p$ has solutions in the set of positive integers if and only if the equation $x^{m'}+y^{n'}=z^{p'}$ has solutions in the set of positive integers. [i]Luminița Popescu[/i]

The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum? $ \textbf{(A)} \ 5 \qquad \textbf{(B)} \ 20 \qquad \textbf{(C)} \ 25 \qquad \textbf{(D)} \ 30 \qquad \textbf{(E)} \ 36$

The circle inscribed in the triangle $ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertex $A$ at the points $Q$ and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles $BKQ$ and $CPL$ lies on $BC$.

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

Three players $A, B$ and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, $....$, $A$ starts the game and the one who takes the last stone loses. Players $A$ and $C$ They form a team against $B$, they agree on a strategy joint. $B$ can take $1, 2, 3, 4$ or $5$ stones on each move, while that $A$ and $C$ can each draw $1, 2$ or $3$ stones in each turn. Determine for which values of $N$ have winning strategies $A$ and $C$ , and for what values the winning strategy is $B$'s.

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]

Let $\{a_n\}$ be a sequence of positive integers such that $a_{n+1} = a_n^2+1$ for all $n \geq 1$. Prove that there is no positive integer $N$ such that $$\prod_{k=1}^N(a_k^2+a_k+1)$$ is a perfect square.

In year $N$, the $300^\text{th}$ day of the year is a Tuesday. In year $N+1$, the $200^\text{th}$ day is also a Tuesday. On what day of the week did the $100^\text{th}$ of year $N-1$ occur? $\text{(A)}\ \text{Thursday}\qquad\text{(B)}\ \text{Friday}\qquad\text{(C)}\ \text{Saturday}\qquad\text{(D)}\ \text{Sunday}\qquad\text{(E)}\ \text{Monday}$

Triangle $ABC$ is isosceles with $AC=BC=25$ and $AB=10$. Let $O$ be the orthocenter of $\triangle ABC$, the intersection of the three altitudes of $\triangle ABC$. Reflect $O$ across $AB$ to a point $D$, and extend $CB$ and $AD$ to intersect at point $E$. Compute the area of $\triangle ABE$.

Let $A\subset \{x|0\le x<1\}$ with the following properties: 1. $A$ has at least 4 members. 2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds. Prove: $A$ has infinetly many members.

Prove that there exists $1904$-element subset of the set $\{1,2,\ldots,1992\}$, which doesn’t contain an arithmetic progression consisting of $41$ terms. [i](Ivan Tonov)[/i]

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

Let an even positive integer $2k$ be given. Find such relatively prime positive integers $x, y$ that maximize the product $xy$.

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$? $\textbf{(A) }6\sqrt 3+3\qquad \textbf{(B) }\dfrac{27}2\qquad \textbf{(C) }8\sqrt 3\qquad \textbf{(D) }14\qquad \textbf{(E) }3\sqrt 3+9\qquad$

[b]p1.[/b] Replace $*$’s by an arithmetic operations (addition, subtraction, multiplication or division) to obtain true equality $$2*0*1*6*7=1.$$ [b]p2.[/b] The interval of length $88$ cm is divided into three unequal parts. The distance between middle points of the left and right parts is $46$ cm. Find the length of the middle part. [b]p3.[/b] A $5\times 6$ rectangle is drawn on a square grid. Paint some cells of the rectangle in such a way that every $3\times 2$ sub‐rectangle has exactly two cells painted. [b]p4.[/b] There are $8$ similar coins. $5$ of them are counterfeit. A detector can analyze any set of coins and show if there are counterfeit coins in this set. The detector neither determines which coins nare counterfeit nor how many counterfeit coins are there. How to run the detector twice to find for sure at least one counterfeit coin? [b]p5.[/b] There is a set of $20$ weights of masses $1, 2, 3,...$ and $20$ grams. Can one divide this set into three groups of equal total masses? [b]p6.[/b] Replace letters $A,B,C,D,E,F,G$ by the digits $0,1,...,9$ to get true equality $AB+CD=EF * EG$ (different letters correspond to different digits, same letter means the same digit, $AB$, $CD$, $EF$, and $EG$ are two‐digit numbers). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ be an irreducible monic polynomial with integer coefficients such that $f(0)$ is not equal to $1$. Let $z$ be a complex number that is a root of $f$. Show that if $w$ is another complex root of $f$, then $\frac{z}{w}$ cannot be a positive integer greater than $1$.

Prove the inequality $\left(1+\frac{1}{q}\right)\left(1+\frac{1}{q^2}\right)...\left(1+\frac{1}{q^n}\right)<\frac{q-1}{q-2}$ for $n\in N, q>2$

[b]p1.[/b] If the pairwise sums of the three numbers $x$, $y$, and $z$ are $22$, $26$, and $28$, what is $x + y + z$? [b]p2.[/b] Suhas draws a quadrilateral with side lengths $7$, $15$, $20$, and $24$ in some order such that the quadrilateral has two opposite right angles. Find the area of the quadrilateral. [b]p3.[/b] Let $(n)*$ denote the sum of the digits of $n$. Find the value of $((((985^{998})*)*)*)*$. [b]p4.[/b] Everyone wants to know Andy's locker combination because there is a golden ticket inside. His locker combination consists of 4 non-zero digits that sum to an even number. Find the number of possible locker combinations that Andy's locker can have. [b]p5.[/b] In triangle $ABC$, $\angle ABC = 3\angle ACB$. If $AB = 4$ and $AC = 5$, compute the length of $BC$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A subset $S$ of positive real numbers is called [i]powerful[/i] if for any two distinct elements $a, b$ of $S$, at least one of $a^{b}$ or $b^{a}$ is also an element of $S$. [b]a)[/b] Give an example of a four elements powerful set. [b]b)[/b] Prove that every finite powerful set has at most four elements.

Set $A$ consists of $2016$ positive integers. All prime divisors of these numbers are smaller than $30.$ Prove that there are four distinct numbers $a, b, c$ and $d$ in $A$ such that $abcd$ is a perfect square.