Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.

The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and have area of at least six square units? [asy] import graph; size(7cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; dot((-2,4),linewidth(6pt) + dotstyle); dot((-1,4),linewidth(6pt) + dotstyle); dot((0,4),linewidth(6pt) + dotstyle); dot((1,4),linewidth(6pt) + dotstyle); dot((2,4),linewidth(6pt) + dotstyle); dot((3,4),linewidth(6pt) + dotstyle); dot((4,4),linewidth(6pt) + dotstyle); dot((-3,2),linewidth(6pt) + dotstyle); dot((-2,2),linewidth(6pt) + dotstyle); dot((-1,2),linewidth(6pt) + dotstyle); dot((0,2),linewidth(6pt) + dotstyle); dot((1,2),linewidth(6pt) + dotstyle); dot((2,2),linewidth(6pt) + dotstyle); dot((3,2),linewidth(6pt) + dotstyle); dot((4,2),linewidth(6pt) + dotstyle); dot((5,2),linewidth(6pt) + dotstyle); [/asy]

Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).

p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$. p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ? p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements. p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of ​​the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$. p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.

Polina wrote on the first page of her notebook $n$ different positive integers. On the second page she wrote all pairwise sums of the numbers from the first page, and on the third - absolute values of pairwise differences of number from the second page. After that she kept doing same operations, i.e. on the page $2k$ she wrote all pairwise sums of numbers from page $2k-1$, and on the page $2k+1$ absolute values of differences of numbers from page $2k$. At some moment Polina noticed that there exists a number $M$ such that, no matter how long she does her operations, on every page there are always at most $M$ distinct numbers. What is the biggest $n$ for which it is possible? [i]M. Karpuk[/i]

Call a set of points in the plane $good$ if any three points of the set are the vertices of an isosceles triangle or if they are collinear. Determine all $a)$ 6-element $good$ sets $b)$ 7-element $good$ sets.

Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$? [i]J. Kalina, L. Lempert[/i]

Consider a function \( n \mapsto f(n) \) that satisfies the following conditions: \( f(n) \) is an integer for each \( n \). \( f(0) = 1 \). \( f(n+1) > f(n) + f(n-1) + \cdots + f(0) \) for each \( n = 0, 1, 2, \dots \). Determine the smallest possible value of \( f(2023) \).

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

Let $ABCD$ be a cyclic quadrilateral, $K$ and $N$ be the midpoints of the diagonals and $P$ and $Q$ be points of intersection of the extensions of the opposite sides. Prove that $\angle PKQ + \angle PNQ = 180$. [i]($7$ points)[/i] .

Show that for a triangle with radii of circumcircle and incircle equal to $ R$, $ r$ respectively, the inequality $ R \geq 2r$ holds.

An acute-angled triangle $ABC$ is given. Points $A_1, B_1$ and $C_1$ are taken on its sides $BC, CA$ and $AB$, respectively, such that $\angle B_1A_1C_1 + 2 \angle BAC = 180^o$, $\angle A_1C_1B_1 + 2 \angle ACB = 180^o$, $\angle C_1B_1A_1 + 2 \angle CBA = 180^o$. Find the locus of the centers of the circles inscribed in triangles $A_1B_1C_1$ (all kinds of such triangles are considered).

Chandler wants to buy a $\$500$ dollar mountain bike. For his birthday, his grandparents send him $\$50$, his aunt sends him $\$35$ and his cousin gives him $\$15$. He earns $\$16$ per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $\mathcal F$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq 1$, $y\geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives): \[xf_x + yf_y = xy\ln(xy),\] \[x^2f_{xx} + y^2f_{yy} = xy.\] For each $f\in\mathcal F$, let \[ m(f) = \min_{s\geq 1}\left(f(s+1,s+1) - f(s+1,s)-f(s,s+1) + f(s,s)\right). \] Determine $m(f)$, and show that it is independent of the choice of $f$.

Tomek invited to a remote birthday part $11$ of his friends who will join the meeting one by one. Tomek chose the guests in such a way that, regardless of the order in which they will join, always the newcomer knew at least half of the people already present, including Tomek. Prove that among of invited guests, there is one who knows all of Tom's other $10$ friends. Caution: We assume that if person A knows person $B$, then $B$ also knows $A$. [hide=original wording]Tomek zaprosił na zdalne przyjęcie urodzinowe 11 swoich znajomych, którzy kolejno będą dołączać do spotkania. Tomek dobrał gości w taki sposób, aby niezależnie od kolejności w jakiej będą dołączać, zawsze nowo przybyła osoba znała co najmniej połowę już obecnych osób, wliczając Tomka. Wykaż, że wśród zaproszonych gości istnieje taki, który zna wszystkich pozostałych 10 znajomych Tomka. Uwaga: Przyjmujemy, że jeśli osoba A zna osobę B, to również B zna A.[/hide]

Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number \[ \frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}} \] is an integer?

Yang has the sequence of integers $1, 2, \dots, 2017$. He makes $2016$ [i]swaps[/i] in order, where a swap changes the positions of two integers in the sequence. His goal is to end with $2, 3, \dots, 2017, 1$. How many different sequences of swaps can Yang do to achieve his goal?

Bilbo, Gandalf, and Nitzan play the following game. First, Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo now compiles a string of length $4044$ built from the three letters $a,b,c$. Nitzan looks at the string, chooses one of the three letters $a,b,c$, and removes from the string all instances of the chosen letter. Only then is the string revealed to Gandalf. He must now guess the number Nitzan chose. Can Bilbo and Gandalf work together and come up with a strategy beforehand that will always allow Gandalf to guess Nitzan's number correctly, no matter how he acts?

Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.

For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]

Prove that for all positive real numbers $a,b,c$ the following inequality holds: \[(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7\] and determine all cases of equality. [i]Lucian Petrescu[/i]

An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.

Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.

Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.