Eve picked some apples, each weighing at most $\frac{1}{2}$ pound. Her apples weigh a total of $W$ pounds, where $W > \frac{1}{3}$. Prove that she can place all her apples into $\left\lceil \frac{3W - 1}{2} \right\rceil$ or fewer baskets, each of which holds up to 1 pound of apples. (The apples are not allowed to be cut into pieces.) Note: If $x$ is a real number, then $\lceil x \rceil$ (the ceiling of $x$) is the least integer that is greater than or equal to $x$.

Point-like figures are placed in the vertices of a regular $k$-angle, and then we walk with them. In one step, a piece jumps over another piece, i.e. its new location will be a mirror image of its current location to the current location of another piece. In the case of $k \geq 3$ integers, it is possible to achieve with a series of such steps that the puppets form the vertices of a regular $k$-angle, different in size from the original?

We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$: \[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\] Find the sum of all $a_{2023}$'s possible values.

On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$. [i]Proposed by Milos Milosavljevic[/i]

Solve this equation with $x \in R$: $x^3-3x=\sqrt{x+2}$

Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$. let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$. prove that circumcentre of triangle $BDX$ lies on $\omega$.

Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.

We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.

A point $P$ lies on one of medians of triangle $ABC$ in such a way that $\angle PAB =\angle PBC =\angle PCA$. Prove that there exists a point $Q$ on another median such that $\angle QBA=\angle QCB =\angle QAC$.

Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.

Consider the rectangular parallelepiped $ ABCDA'B'C'D' $ and the point $ O $ to be the intersection of $ AB' $ and $ A'B. $ On the edge $ BC, $ pick a point $ N $ such that the plane formed by the triangle $ B'AN $ has to be parallel to the line $ AC', $ and perpendicular to $ DO'. $ Prove, then, that this parallelepiped is a cube.

Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum $$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.

Prove that for every integer $n \ge 1$ there exists a real number $a$ such that for any integer $m \ge 1$ the number $[a^m] + 1$ is divisible by $n$ ($[x]$ denotes the largest integer that does not exceed $x$).

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Let $B$ be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer 1 is obtained.

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$

An infinite sequence of positive real numbers $x_0,x_1,x_2,...$ is called $vasco$ if it satisfies the following properties: (a) $x_0=1,x_1=3$; and (b) $x_0+x_1+...+x_{n-1}\ge3x_{n}-x_{n+1}$, for every $n\ge1$. Find the greatest real number $M$ such that, for every $vasco$ sequence, the inequality $\frac{x_{n+1}}{x_{n}}>M$ is true for every $n\ge0$.

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

Is the series $$\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}$$ convergent or divergent?

Prove that there are infinitely many quadruplets of positive integers $(a,b,c,d)$, such that\\ $ab+1$, $bc+16$, $cd+4$, $ad+9$\\ are perfect squares

For a positive integer $n$, we will say that a sequence $a_1, a_2, \dots a_n$ where $a_i \in \{1, 2, \dots , n\}$ for all $i$ is $n$[i]-highly divisible[/i] if, for every positive integer $d$ that divides $n$ and every nonnegative integer $k$ less than $\frac{n}{d}$ we have that \[ d\;\Bigg\vert \sum_{i=kd+1}^{(k+1)d} a_i. \] Let $\chi(n)$ be the probability that a sequence $a_1, a_2, \dots, a_n$ where $a_i$ is chosen randomly from $\{1, 2, \dots n\}$ independently for all $i$ is $n$-highly divisible. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ not divisible by 3 such that $3^{40}\chi(n)=\frac{1}{m}$. Compute the sum of all possible values of $n$. [i]Proposed by Jaedon Whyte[/i]

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with \[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\] for all $x,y,z \in \mathbb{R}_{\ge 0}$.