A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players $A$ and $B$. A third person called the [i]dealer[/i] shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the [i]receiver[/i], takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver. $A$ is initially the receiver and takes the first card dealt. The player who first obtains a complete set of $n$ different cards wins the game. What fraction of all possible arrangements of the pack lead to $A$ winning? Prove the correctness of your answer.

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar. [asy] unitsize (0.5 cm); pair A, B, C, D, E, F, G, H; pair x, y, z; x = (1,0.5); y = (-0.8,0.8); z = (0,1); B = (0,0); C = 5*x; A = 3*y; F = 4*z; E = A + F; G = C + F; H = A + C + F; fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8)); fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8)); fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8)); fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8)); draw(B--C--G--H--E--A--cycle); draw(B--F); draw(E--F); draw(G--F); draw(y--(y + 4*z)--(y + 4*z + 5*x)); draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y)); draw((3*y + z)--z--(5*x + z)); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, SE); label("$E$", E, NW); label("$F$", F, SE); label("$G$", G, NE); label("$H$", H, N); [/asy]

Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$. Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a “unique” point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$. (translated by Miklós Maróti)

[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients. [/color]

Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

A word is a sequence of capital letters of our alphabet (that is, there are 26 possible letters). A word is called palindrome if has at least two letters and is spelled the same forward and backward. For example, the words "ARARA" e "NOON" are palindromes, but the words "ESMERALDA" and "A" are not palindromes. We say that a word $x$ contains a word $y$ if there are consecutive letters of $x$ that together form the word $y$. For example, the word "ARARA" contains the word "RARA" and also the word "ARARA", but doesn't contain the word "ARRA". Compute the number of words of 14-letter that contain some palindrome.

Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).

If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$? $\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$

Estimate the number of five-card combinations from a standard $52$-card deck that contain a pair (two cards with the same number). An estimate of $E$ earns $2e^{-\frac{\left|A-E\right|}{20000}}$ points, where $A$ is the actual answer. [i]2018 CCA Math Bonanza Lightning Round #5.1[/i]

Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? $\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

The sum to infinity of the terms of an infinite geometric progression is $ 6$. The sum of the first two terms is $ 4\frac {1}{2}$. The first term of the progression is: $ \textbf{(A)}\ 3 \text{ or } 1\frac {1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2\frac {1}{2} \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 9 \text{ or } 3$

On the coordinate plane, draw all points $M(x, y)$, whose coordinates satisfy the equation: $$ |x-y| + |1-x| + |y|=1 $$

Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$ for all reals $x, y $.

Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds; [*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]

Define the [i]annoyingness[/i] of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2 \ldots ,n$ appears. For instance, the annoyingness of $3,2,1$ is $3,$ and the annoyingness of $1,3,4,2$ is $2.$ A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.

Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides?

Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values ​​of $N $.

A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?

Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$ and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\

$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that: \[ BT+CT\leq{2R}. \]