Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$. Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.

The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$. [asy] size(4cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; draw(unitcircle); pair A,B,C,D,E,F; A=dir(90); B=dir(66); C=dir(18); D=dir(282); E=dir(210); F=origin; dot("$F$",F,NW); dot("$A$",A,dir(90)); dot("$B$",B,dir(66)); dot("$C$",C,dir(18)); dot("$D$",D,dir(306)); dot("$E$",E,dir(210)); draw(F--E^^F--D^^F--C^^F--B^^F--A); [/asy]

Let $n$ and $k$ be positive integers. An $n$-tuple $(a_1, a_2,\ldots , a_n)$ is called a permutation if every number from the set $\{1, 2, . . . , n\}$ occurs in it exactly once. For a permutation $(p_1, p_2, . . . , p_n)$, we define its $k$-mutation to be the $n$-tuple $$(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),$$ where indices are taken modulo $n$. Find all pairs $(n, k)$ such that every two distinct permutations have distinct $k$-mutations. [i]Remark[/i]: For example, when $(n, k) = (4, 2)$, the $2$-mutation of $(1, 2, 4, 3)$ is $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$. [i]Proposed by Borna Šimić[/i]

Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$ Here $d(n)$ is the number of positive divisors of a positive integer

Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex. a) In how many ways can the ant walk around each vertex exactly twice? b) In how many ways can the ant walk around each vertex exactly three times? Note: For each item, consider that the starting vertex is visited.

A string consisting of letters $A, C, G,$ and $U$ is [i]untranslatable[/i] if and only if it has no $\text{AUG}$ as a consecutive substring. For example, $\text{ACUGG}$ is untranslatable. Let $a_n$ denote the number of untranslatable strings of length $n.$ It is given that there exists a unique triple of real numbers $(x,y,z)$ such that $a_n = xa_{n-1} + ya_{n-2} +za_{n-3}$ for all integers $n \ge 100.$ Compute $(x, y,z)$

The numbers $A_1 , A_2 ,... , A_n$ are given. Prove, without calculating derivatives, that the value of $X$ that minimizes the sum $(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2$ is precisely the arithmetic mean of the given numbers.

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.