Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

Prove that inside any acute-angled triangle, there exists a point $P$ such that the feet of the perpendiculars dropped from $P$ to the sides of the triangle are the vertices of an equilateral triangle. (NB Vassiliev)

Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

In an acute triangle $ABC$, its inscribed circle touches the sides $AB,BC$ at the points $C_1,A_1$ respectively. Let $M$ be the midpoint of the side $AC$, $N$ be the midpoint of the arc $ABC$ on the circumcircle of triangle $ABC$, and $P$ be the projection of $M$ on the segment $A_1C_1$. Prove that the points $P,N$ and the incenter $I$ of the triangle $ABC$ lie on the same line.

$\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$ $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{2}{3} \qquad \text{(C)}\ \dfrac{3}{2} \qquad \text{(D)}\ \dfrac{17}{3} \qquad \text{(E)}\ \dfrac{34}{3}$

Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied: 1- $b>a$ and $b-a$ is a prime number 2- The last digit of the number $a+b$ is $3$ 3- The number $ab$ is a square of an integer.

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

Given a real number $\alpha$ and consider function $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$. Find all function $f:\mathbb R\to\mathbb R$ that satisfy: $$f(\varphi(x)+f(y))=y+\varphi(f(x))$$ forall $x,y\in\mathbb R$

Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

Let $D$ be the midpoint of the base $BC$ of an isosceles triangle $ABC,$ $E$ be the point at the side $AC$ such that $\angle CDE=60^\circ,$ and $M$ be the midpoint of $DE.$ Prove that $\angle AME=\angle BMD.$

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans? $\textbf{(A) } 1:4 \qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1$

Let $X \subset \mathbb{R}^2$ be a set satisfying the following properties: (i) if $(x_1,y_1)$ and $(x_2,y_2)$ are any two distinct elements in $X$, then \[\text{ either, }\ \ x_1>x_2 \text{ and } y_1>y_2\\ \text{ or, } \ \ x_1<x_2 \text{ and } y_1<y_2\] (ii) there are two elements $(a_1,b_1)$ and $(a_2,b_2)$ in $X$ such that for any $(x,y) \in X$, \[a_1\le x \le a_2 \text{ and } b_1\le y \le b_2\] (iii) if $(x_1,y_1)$ and $(x_2,y_2)$ are two elements of $X$, then for all $\lambda \in [0,1]$, \[\left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X\] Show that if $(x,y) \in X$, then for some $\lambda \in [0,1]$, \[x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2\]

Show that for any real $x\in[0,1]$ the inequality \[\frac{(1-x)x^2}{(1+x)^3}<\frac{1}{25}\] holds.

Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

Let $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1}, \ldots, a_1$). As an example, let $A$ be the sequence $1,1,0,0,1$. We can pick block $1,0,0$ and flip it, so the sequence $1,\boxed{1,0,0},1$ becomes $1,\boxed{0,0,1},1$. However, we cannot pick block $1,1,0,0$ and flip their order since they contain the same number of $1$s and $0$s. Two sequences $A$ and $B$ are called [i]related[/i] if $A$ can be transformed into $B$ using a finite number the operation mentioned above. Determine the largest natural number $n$ for which there exists $n$ different sequences $A_1, A_2, \ldots, A_n$ where each sequence consists of 2022 digits, and for every index $i \neq j$, the sequence $A_i$ is not related to $A_j$.

The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.

Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.

Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that \[a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) \]

Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows: $\text{I.}$ They play alternately. $\text{II.}$ A turn consists of placing one checker on an unoccupied square of the board. $\text{III.}$ In her turn, a player may select either a white checker or a black checker and need not always use the same colour. $\text{IV.}$ When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers. $\text{V.}$ The player obtaining at least five of the eight points WINS. $\text{(a)}$ Is a $4-4$ tie possible? Explain. $\text{(b)}$ Describe a winning strategy for the girl who is first to play.

An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$. What is the area of triangle $ABC$?

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

Let $I$ be the incenter of a triangle $ABC$ and let $A', B', C'$ be midpoints of sides $BC$, $CA$, $AB$, respectively. If $IA'= IB'= IC'$ , then prove that triangle $ABC$ is equilateral.

On a $ 30 \times30 $ square board or placed figures of shape 1 (of 5 squares) (in all four possible positions) and shaped figures of shape 2 (of 4 squares) . The figures do not overlap, they do not pass through the edges of the board and the squares of which they are drawn lie exactly through the squares of the board. a) Prove that the board can be fully covered using $100$ figures of both shapes. b) Prove that if there are already $50$ shaped figures on the board of shape 1, then at least one more figure can be placed on the board. c) Prove that if there are already $28$ figures of both shapes on the board then at least one more figure of both shapes can be placed on the board. [img]https://cdn.artofproblemsolving.com/attachments/3/f/f20d5a91d61557156edf203ff43acac461d9df.png[/img]