Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that: $a)$ $J$ is an interval; $b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements; $c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.

Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle ABP=45^\circ$. Given that $AP=1$ and $CP=2$, compute the area of $ABC$.

Let $n$ be a positive integer. Given $n^2$ points in a unit square, prove that there exists a broken line of length $2n + 1$ that passes through all the points. [i]Allen Yuan[/i]

Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.

Find all pairs of positive integers $(a,b)$ such that there exists a finite set $S$ satisfying that any positive integer can be written in the form $$n = x^a + y^b + s$$where $x,y$ are nonnegative integers and $s \in S$ [i]CSJL[/i]

In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. Thereafter, $7$ out of every $8$ counted were red. If, in all, $90\%$ or more of the balls counted were red, the maximum value of $n$ is: $\textbf{(A)}\ 225 \qquad \textbf{(B)}\ 210 \qquad \textbf{(C)}\ 200 \qquad \textbf{(D)}\ 180 \qquad \textbf{(E)}\ 175$

Find the critical values and critical points of the mapping $z\mapsto z^2+2\overline{z}$ (sketch the answer).

In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 2 \sqrt {3}\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 4 \sqrt {2}$

Prove that there don't exist positive integer numbers $k$ and $n$ which satisfy equation $n^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k$. [i]Proposed by Mykhailo Shtandenko[/i]

Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have? $\textbf{(A) }8062 \qquad \textbf{(B) }8068 \qquad \textbf{(C) }8072 \qquad \textbf{(D) }8076 \qquad \textbf{(E) }8082$

The pentagon $ABCDE$ is inscribed in the circle $\omega$. Let $T$ be the intersection point of the diagonals $BE$ and $AD$. A line is drawn through the point $T$ parallel to $CD$, which intersects $AB$ and $CE$ at points $X$ and $Y$, respectively. Prove that the circumscribed circle of the triangle $AXY$ is tangent to $\omega$.

The expression $ 1 \minus{} \frac {1}{1 \plus{} \sqrt {3}} \plus{} \frac {1}{1 \minus{} \sqrt {3}}$ equals: $ \textbf{(A)}\ 1 \minus{} \sqrt {3} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \minus{} \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3} \qquad\textbf{(E)}\ 1 \plus{} \sqrt {3}$

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that: $ \text{(i)}\quad f(0)=0 $ $ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $ $ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $ Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$ is primitivable.

In triangles $ABC$ and $DEF$, lengths $AC,~BC,~DF,$ and $EF$ are all equal. Length $AB$ is twice the length of the altitude of $\triangle DEF$ from $F$ to $DE$. Which of the following statements is (are) true? $\textbf{I. }\angle ACB \text{ and }\angle DFE\text{ must be complementary.}$ $\textbf{II. }\angle ACB \text{ and }\angle DFE\text{ must be supplementary.}$ ${\textbf{III. }\text{The area of }\triangle ABC\text{ must equal the area of }\triangle DEF.}$ ${\textbf{IV. }\text{The area of }\triangle ABC\text{ must equal twice the area of }\triangle DEF.}$ $\textbf{(A) }\textbf{II. }\text{only}\qquad\textbf{(B) }\textbf{III. }\text{only}\qquad$ $\textbf{(C) }\textbf{IV. }\text{only}\qquad\textbf{(D) }\text{I. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\textbf{II. }\text{and }\textbf{III. }\text{only}$

In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that (i) every student goes to the same destination as their best friend; (ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to $1$.

Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely? $ \textbf{(A) }\text{all 4 are boys}$\\ $\textbf{(B) }\text{all 4 are girls}$\\$ \textbf{(C) }\text{2 are girls and 2 are boys}$\\ $\textbf{(D) }\text{3 are of one gender and 1 is of the other gender}$\\ $\textbf{(E) }\text{all of these outcomes are equally likely} $

Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? $\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that: $P(x) = b + (x-a)Q((x-a)^2)).$

Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$). a) Prove that $\angle{BAM}=\angle{CAN}$. b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.

Angle $ B$ of triangle $ ABC$ is trisected by $ BD$ and $ BE$ which meet $ AC$ at $ D$ and $ E$ respectively. Then: $ \textbf{(A)}\ \frac {AD}{EC} \equal{} \frac {AE}{DC} \qquad\textbf{(B)}\ \frac {AD}{EC} \equal{} \frac {AB}{BC} \qquad\textbf{(C)}\ \frac {AD}{EC} \equal{} \frac {BD}{BE}$ $ \textbf{(D)}\ \frac {AD}{EC} \equal{} \frac {AB\cdot BD}{BE\cdot BC} \qquad\textbf{(E)}\ \frac {AD}{EC} \equal{} \frac {AE\cdot BD}{DC\cdot BE}$

It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to $3$ edges.) [i]($10$ points)[/i]