A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.

For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2024}{n} \right]$. Find the value of $$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Anna is out shopping for fruit. She observes that four oranges, three bananas and one lemon costs exactly the same as three oranges and two lemons (all prices are in whole kroner). Just then her friend Bengt calls, and Anna tells this to him. Bengt complains, that ''information is not enough for me to know how much each fruit costs''. ''No'', says Anna,' 'but three oranges and two lemons cost as many kroner as your mother is old''. Unfortunately, it's not enough either, but if she had been younger then the information would have been sufficient for you to be able to figure out what the fruits costs. How old is Bengt's mother?

Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

Find all positive integers $ m$ with the property that the fourth power of the number of (positive) divisors of $ m$ equals $ m$.

The triangle below is divided into nine stripes of equal width each parallel to the base of the triangle. The darkened stripes have a total area of $135$. Find the total area of the light colored stripes. [img]https://cdn.artofproblemsolving.com/attachments/0/8/f34b86ccf50ef3944f5fbfd615a68607f4fadc.png[/img]

Let $T$ be an arbitrary tetrahedron satisfying the following conditions: [b]I.[/b] Each its side has length not greater than 1, [b]II.[/b] Each of its faces is a right triangle. Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.

A fifth number,$n$ , is added to the set $ \{ 3,6,9,10\} $ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ \text{more than }4 $

Nair and Yuli play the following game: $1.$ There is a coin to be moved along a horizontal array with $203$ cells. $2.$ At the beginning, the coin is at the first cell, counting from left to right. $3.$ Nair plays first. $4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right. $5.$ The winner is the one who reaches the last cell first. What strategy does Nair need to use in order to always win the game?

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color. [b]a.)[/b] Show that: [i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue. [i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.) [b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

Given triangle $ABC$, let $D$ be an inner point of segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC, L=CP\cap AB, E=BQ\cap AC, F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.

In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.

For every positive integer $n$, $f(2n+1)=2f(2n)$, $f(2n)=f(2n-1)+1$, and $f(1)=0$. What is the remainder when $f(2005)$ is divided by $5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.

Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that [b](i)[/b] $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and [b](ii)[/b] $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.

$7662$ chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either $1$st, $2$nd, or $3$rd grade students, so that there are $2554$ chairs labeled with each label. The following situations happen, in order [list=i] [*] $2554$ students each from the $1$st, $2$nd, and $3$rd grades are given a ball as follows: $1$st grade students receive footballs, $2$nd grade students receive basketballs, and $3$rd grade students receive volleyballs. [*] The students go sit in chairs labeled for their grade [*] The students simultaneously send their balls to the student to their left, and this happens some positive number of times. [/list] A labelling of the chairs is called [i]lin-ping[/i] if it is possible for all $1$st, $2$nd, and $3$rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of [i]lin-ping[/i] labellings

Determine all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying the condition $ f(xy) \le xf(y)$ for all real numbers $ x$ and $ y$.

Betty has a $3\times 4$ grid of dots. She colors each dot either red or maroon. Compute the number of ways Betty can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color.

The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.

Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$

On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs?