Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$? [hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide]

Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $. $\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $. $\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $. Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.

On a blank piece of paper, two points with distance $1$ is given. Prove that one can use (only) straightedge and compass to construct on this paper a straight line, and two points on it whose distance is $\sqrt{2021}$ such that, in the process of constructing it, the total number of circles or straight lines drawn is at most $10.$ Remark: Explicit steps of the construction should be given. Label the circles and straight lines in the order that they appear. Partial credit may be awarded depending on the total number of circles/lines.

Kongol rolls two fair 6-sided die. The probability that one roll is a divisor of the other can be expressed as $\frac{p}{q}$. Determine $p+q$. [i]2022 CCA Math Bonanza Lightning Round 3.1[/i]

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$. Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum? Personal note: They really phoned it in with the names there…

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .

A point $P$ is given inside an acute-angled triangle $ABC$. Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

Express $2.3\overline{57}$ as a common fraction. [i]2017 CCA Math Bonanza Lightning Round #3.1[/i]

Determine the set of all real numbers $p$ for which the polynomial $Q(x) = x^3 + px^2 - px - 1$ has three distinct real roots.

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

find the smallest integer $n\geq1$ such that the equation : $$a^2+b^2+c^2-nd^2=0 $$ has $(0,0,0,0)$ as unique solution .

Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.

a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$ We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$ Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$ b) Prove that the product of $2$ original polynomials is a original polynomial.

Consider these two geoboard quadrilaterals. Which of the following statements is true? [asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 5; ++b) { dot((a,b)); } } draw((0,3)--(0,4)--(1,3)--(1,2)--cycle); draw((2,1)--(4,2)--(3,1)--(3,0)--cycle); label("I",(0.4,3),E); label("II",(2.9,1),W); [/asy] $\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}$ $\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}$ $\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}$ $\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$ $\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$

You have the following pieces: one $4\times 1$ rectangle, two $3\times 1$ rectangles, three $2\times 1$ rectangles, and four $1\times 1$ squares. Ariel and Bernardo play the following game on a board of $n\times n$, where $n$ is a number that Ariel chooses. In each move, Bernardo receives a piece $R$ from Ariel. Next, Bernardo analyzes if he can place $R$ on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for $R$, Bernardo must choose one of them and place $R$. The game stops if it is impossible to place $R$ in the way explained, and Bernardo wins. Ariel wins only if all $10$ pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the $n$ found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory? Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

Cheryl rolls a fair dice twice. If the dice's six faces are numbered by $1$, $2$, $3$, $4$, $5$, $6$, what is the probability that the number on one of her rolls is a divisor of the number on the other roll? $\textbf{(A) }\dfrac29\qquad\textbf{(B) }\dfrac5{18}\qquad\textbf{(C) }\dfrac49\qquad\textbf{(D) }\dfrac12\qquad\textbf{(E) }\dfrac{11}{18}$

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$? $\textbf{(A) }10\qquad\textbf{(B) }18\qquad\textbf{(C) }25\qquad\textbf{(D) }36\qquad\textbf{(E) }81$