Let $P_1,...,P_n$ and $Q_1,...,Q_n$ be oppositely oriented convex polygons. Prove that there is a line passing through the n line segments $P_1Q_1,...,P_nQ_n$.

We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.) What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.

If $ 200 \le a \le 400$ and $ 600 \le b \le 1200$, then the largest value of the quotient $ \frac{b}{a}$ is \[ \textbf{(A)}\ \frac{3}{2} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 300 \qquad \textbf{(E)}\ 600 \qquad \]

A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is: $\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ \textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ \textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ \textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ \textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$

Decide, whether exist tetrahedrons $T$, $T'$ with walls $S_1$, $S_2$, $S_3$, $S_4$ and $S_1'$, $S_2'$, $S_3'$, $S_4'$, respectively, such that for $i = 1, 2, 3, 4$ triangle $S_i$ is similar to triangle $S_i'$, but despite this, tetrahedron $T$ is not similar to tetrahedron $T'$.

Let $N$ be an integer greater than $1$. A deck has $N^3$ cards, each card has one of $N$ colors, has one of $N$ figures and has one of $N$ numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

We call a set $S$ a [i]good[/i] set if $S=\{x,2x,3x\}(x\neq 0).$ For a given integer $n(n\geq 3),$ determine the largest possible number of the [i]good[/i] subsets of a set containing $n$ positive integers.

In a box of $100$ marbles, just $3\%$ of the marbles are purple and the rest are green. How many green marbles must be removed from the box so that $95\%$ of the remaining marbles are green? $\text{(A) }2\qquad\text{(B) }15\qquad\text{(C) }37\qquad\text{(D) }40\qquad\text{(E) }57$

The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square? (A Tolpygo)

Let $a_{1}, \cdots, a_{44}$ be natural numbers such that \[0<a_{1}<a_{2}< \cdots < a_{44}<125.\] Prove that at least one of the $43$ differences $d_{j}=a_{j+1}-a_{j}$ occurs at least $10$ times.

Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$ for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$. [i]Proposed by Navilarekallu Tejaswi[/i]

How many terms are there in the arithmetic sequence $13, 16, 19, \dots, 70,73$? $ \textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }24\qquad\textbf{(D) }60\qquad\textbf{(E) }61 $

There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$.

Find all triples \( (x, y, z) \) of positive integers that satisfy the equation \[ x + xy + xyz = 31. \]

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? $ \textbf{(A)}\ 56 \qquad \textbf{(B)}\ 58 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 64$

Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?

Find all quadruples $(a, b, c, d)$ of positive integers satisfying $\gcd(a, b, c, d) = 1$ and \[ a | b + c, ~ b | c + d, ~ c | d + a, ~ d | a + b. \] [i]Vítězslav Kala (Czech Republic)[/i]

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.

Let $ABC$ be a triangle such that $|AB|=13 , |BC|=12$ and $|CA|=5$. Let the angle bisectors of $A$ and $B$ intersect at $I$ and meet the opposing sides at $D$ and $E$, respectively. The line passing through $I$ and the midpoint of $[DE]$ meets $[AB]$ at $F$. What is $|AF|$? $ \textbf{(A)}\ \dfrac{3}{2} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \dfrac{5}{2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \dfrac{7}{2} $

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$? $\textbf{(A) }-2\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6\qquad$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$. [i]George Xing.[/i]

There are two imposters and seven crewmates on Polus. How many ways are there for the nine people to split into three groups of three, such that each group has at least two crewmates? Assume that the two imposters and seven crewmates are all distinguishable from each other, but that the three groups are not distinguishable from each other.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.