For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$), a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$) in a way such that $AM -BC = BN- AC = CP – AB$. Prove that the angles of triangle $MNP$ do not depend on the choice of $M, N, P$ .

As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all. Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?" "Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines. Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.

Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true? $\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$

In how many ways can the sum of 100 fillér be made up with coins of denominations l, 2, 10, 20 and 50 fillér?

The bisector of $ \angle BAC $ of a triangle $ ABC $ meet the segment $ BC $ at $ D. $ Through the midpoint of $ AD $ passes aline that intersects $ AB,AC $ at $ M,N, $ respectively. Show that: $$ \frac{1}{MA}+\frac{1}{NA} =2\left( \frac{1}{AB} +\frac{1}{AC} \right) $$ [i]Toni Mihalcea[/i]

Prove that for any positive integer $n$ there exist coprime numbers $a$ and $b$ such that for all $1 \leq k \leq n$ numbers $a+k$ and $b+k$ are not coprime.

Let $f(x) = x^2 + \lfloor x\rfloor ^2 - 2x \lfloor x \rfloor + 1$. Compute $f\left(4 + \frac56 \right)$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3 \rfloor = 3$ and $\lfloor - 4.25 \rfloor = -5$.

On the sides of triangle \( ABC \), points \( D_1, D_2, E_1, E_2, F_1, F_2 \) are chosen such that when going around the triangle, the points occur in the order \( A, F_1, F_2, B, D_1, D_2, C, E_1, E_2 \). It is given that \[ AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2. \] Prove that the perimeters of the triangles formed by the lines \( AD_1, BE_1, CF_1 \) and \( AD_2, BE_2, CF_2 \) are equal.

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$, whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$, there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$-th person is moving $i_n$ places in the right, all people will sit on different places. [i]V. Drenski[/i]

Compute the minimum value of $$\frac{x^4+2x^3+3x^2+2x+10}{x^2+x+1}$$ where $x$ can be any real number.

Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold: $a)$ $f(mn) = f(m)f(n)$ $b)$ $f(m^2 + n^2) \mid  f(m^2) + f(n^2).$

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $ABC$ be a triangle. Points $D, E, F$ are on segments $BC$, $CA$, $AB$, respectively. Suppose that $AF = 10$, $F B = 10$, $BD = 12$, $DC = 17$, $CE = 11$, and $EA = 10$. Suppose that the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect again at $X$. Find the circumradius of $\vartriangle EXF$.

Petya and Kolya play the following game: they take turns changing one of the coefficients $a$ or $b$ of the quadratic trinomial $f = x^2 + ax + b$: Petya is on $1$, Kolya is on $1$ or $3$. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds $a$ and $b$ regardless of Petya's game? [hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?[/hide]

[b]a)[/b] Prove that if $G$ is $2$-connected, then it has a cycle with the length at least $\min\{n(G),2\delta(G)\}$. (10 points) [b]b)[/b] Prove that every $2k$-regular graph with $4k+1$ vertices has a hamiltonian cycle. (10 points)

Find the number of positive and negative squares in the canonical form of the quadratic form $\sum_{i<j}(x_i-x_j)^2$ in $n$ variables. The same for the form $\sum_{i<j}x_i x_j$.

What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times ... \times 50$ is divisible by $10^n$?

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$

The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.

Justify the statement that $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\dots}}}}}.$$

The sequence $ (a_n)$ is defined by $ a_1\equal{}a_2\equal{}a_3\equal{}1$ and $ a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2$ for all $ n \ge 3.$ Prove that $ a_n$ is a positive integer for all $ n \ge 1$.

Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}\plus{}\sqrt{5})^{2000}.$

For postive integers $k,n$, let $$f_k(n)=\sum_{m\mid n,m>0}m^k$$ Find all pairs of positive integer $(a,b)$ such that $f_a(n)\mid f_b(n)$ for every positive integer $n$.

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.