Given triangle $ABC$. Point $B_1$ bisects the length of the broken line $ABC$ (composed of segments $AB$ and $BC$), point $C_1$ bisects the length of the broken line$ACB$, point $A_1$ bisects the length of of the broken line $CAB$. Through points $A_1$, $B_1$ and $C_1$ straight lines $\ell_A$ ,$\ell_B$, $\ell_C$ are drawn parallel to the bisectors angles $BAC$, $ABC$ and $ACB$ respectively. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ intersect at one point.

In a right triangle $ABC$ with $\angle BAC =90^o $and $\angle ABC= 54^o$, point $M$ is the midpoint of the hypotenuse $[BC]$ , point $D$ is the foot of the angle bisector drawn from the vertex $C$ and $AM \cap CD = \{E\}$. Prove that $AB= CE$.

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

The Fibonacci sequence is dened by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that \[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\] If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$.

Let $M=\{3,4,5,6,7,8,9,10,11,12,13,15,17,19,21,23\}$. Prove that for any $a_i>0,i=\overline{1,n},n\in M$ the inequality holds: $$\frac{a_1^2}{(a_2+a_3)^4}+\frac{a_2^2}{(a_3+a_4)^4}+\ldots+\frac{a_{n-1}^2}{(a_n+a_1)^4}+\frac{a_n^2}{(a_1+a_2)^4}\ge\frac{n^3}{16s^2},$$ where $s=\sum_{i=1}^na_i$. [i]Proposed by Marius Olteanu[/i]

Inside triangle \( ABC \), points \( D \) and \( E \) are chosen such that \( \angle ABD = \angle CBE \) and \( \angle ACD = \angle BCE \). Point \( F \) on side \( AB \) is such that \( DF \parallel AC \), and point \( G \) on side \( AC \) is such that \( EG \parallel AB \). Prove that \( \angle BFG = \angle BDC \). [i]Proposed by Anton Trygub[/i]

If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.

Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$.

Suppose that $ f: [a,b]\rightarrow \mathbb{R} $ be a monotonic function and for every $ x_1,x_2\in [a,b] $ that $ x_1<x_2 $ ,there exist $ c\in (a,b) $ such that $ \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) $ a) Show that $ f $ be the continuous function on interval $ (a,b) $ b) Suppose that $ f $ is integrable function on interval $ [a,b] $ but $ f $ isn't a monotonic function then ,is it the result of part a) right?

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

The bisector of acute angle $\alpha$ of the right triangle $ABC$ splits the side $a$ in two segments with lengths $m$ and $n$. Find the acute angle $\beta$ and the lenths of the other two sides by knowing $m$ and $n$.

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

Two circles with radii $3$ and $4$ are externally tangent at $P$. Let $A \neq P$ be on the first circle and $B \neq P$ be on the second circle, and let the tangents at $A$ and $B$ to the respective circles intersect at $Q$. Given that $QA = QB$ and $AB$ bisects $PQ$, compute the area of $QAB$.

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the feet of the altitudes on the sides $BC$ or $AC$. Points $F$ and $G$ are located on the lines $AD$ and $BE$ in such a way that$ \frac{AF}{FD}=\frac{BG}{GE}$. The line passing through $C$ and $F$ intersects $BE$ at point $H$, and the line passing through $C$ and $G$ intersects $AD$ at point $I$. Prove that points $F, G, H$ and $I$ lie on a circle. (Walther Janous)

[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$ [b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$. [b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)? [b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$? [b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$? [b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ : [b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$. [b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS? [b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$. [b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$? [b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$? [b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball? [b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$. [i]Proposed by Ankan Bhattacharya[/i]

What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

Let $n$ be a positive integer. We know that the set $I_n = \{ 1, 2,\ldots , n\}$ has exactly $2^n$ subsets, so there are $8^n$ ordered triples $(A, B, C)$, where $A, B$, and $C$ are subsets of $I_n$. For each of these triples we consider the number $\mid A \cap B \cap C\mid$. Prove that the sum of the $8^n$ numbers considered is a multiple of $n$. Clarification: $\mid Y\mid$ denotes the number of elements in the set $Y$.

$480$ $1$ cm unit cubes are used to build a block measuring $6$ cm by $8$ cm by $10$ cm. A tiny ant then chews his way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that he does not "pass through" cubes if he is merely passing through where their edges or vertices meet. [i]2017 CCA Math Bonanza Individual Round #11[/i]

[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$ [b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$. [b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$. [b]p4.[/b] Find the irrational number: $$A =\sqrt{ \frac12+\frac12 \sqrt{\frac12+\frac12 \sqrt{ \frac12 +...+ \frac12 \sqrt{ \frac12}}}}$$ ($n$ square roots). [b]p5.[/b] The Math country has the shape of a regular polygon with $N$ vertexes. $N$ airports are located on the vertexes of that polygon, one airport on each vertex. The Math Airlines company decided to build $K$ additional new airports inside the polygon. However the company has the following policies: (i) it does not allow three airports to lie on a straight line, (ii) any new airport with any two old airports should form an isosceles triangle. How many airports can be added to the original $N$? [b]p6.[/b] The area of the union of the $n$ circles is greater than $9$ m$^2$(some circles may have non-empty intersections). Is it possible to choose from these $n$ circles some number of non-intersecting circles with total area greater than $1$ m$^2$? PS. You should use hide for answers.