Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves \[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \] partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)

Consider the circles $\omega$, $\omega_1$, $\omega_2$, where $\omega_1$, $\omega_2$ pass through the center $O$ of $\omega$. The circle $\omega$ cuts $\omega_1$ at $A, E$ and $\omega_2$ at $C, D$. The circles $\omega_1$ and $\omega_2$ intersect at $O$ and $M$. If A$D$ cuts $CE$ at $B$ and if $MN \perp BO$, ($N \in BO$) prove that $2MN^2 \le BM \cdot MO$.

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

What is the smallest positive integer $n$ such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true? [i]2019 CCA Math Bonanza Team Round #5[/i]

5. Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25. Find the distance from T to D.

Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that (a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence); (b) if the sequence is periodic, then the first number is rational. (G Shabat)

At the two ends $A, B$ of a diameter (of length $2r$) of a pavement horizontal circular rise two vertical columns, of equal height h, whose ends support a beam $A' B' $ of length equal to the before mentioned diameter. It forms a covered by placing numerous taut cables (which are admitted to be rectilinear), joining points of the beam $A'B'$ with points of the circumference edge of the pavement, so that the cables are perpendicular to the beam $A'B'$ . You want to find out the volume enclosed between the roof and the pavement. [hide=original wording]En los dos extremos A, B de un di´ametro (de longitud 2r) de un pavimento circular horizontal se levantan sendas columnas verticales, de igual altura h, cuyos extremos soportan una viga A' B' de longitud igual al diametro citado. Se forma una cubierta colocando numerosos cables tensos (que se admite que quedan rectilıneos), uniendo puntos de la viga A'B' con puntos de la circunferencia borde del pavimento, de manera que los cables queden perpendiculares a la viga A'B' . Se desea averiguar el volumen encerrado entre la cubierta y el pavimento.[/hide]

Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that $$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$ for all $x,y \in \mathbb{R}$

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. The aquarium is filled with water to a depth of $ 37$ cm. A rock with volume $ 1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? $ \textbf{(A)}\ 0.25 \qquad \textbf{(B)}\ 0.5 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 2.5$

[i]Triangular numbers[/i] are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$, that is $1, 3, 6, 10$, . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers. [i]Proposed by A. Golovanov[/i]

Calculate the limit of the following sequences: [b]a)[/b] n^{n!}/(n!)^n [b]b)[/b] n^{ln n}/n! [i]Adrian Troie[/i]

In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

Fix a positive integer $n$ and a finite graph with at least one edge; the endpoints of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range from $0$ to $n-1$, one number each. A vertex assignment and an edge assignment are [i]compatible[/i] if the following condition is satisfied at each vertex $v$: The number assigned to $v$ is congruent modulo $n$ to the sum of the numbers assigned to the edges incident to $v$. Fix a vertex assignment and let $N$ be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment. Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$.

Given a natural number $k$, find the smallest natural number $C$ such that $$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$.

Lizzie writes a list of fractions as follows. First, she writes $\frac11$, the only fraction whose numerator and denominator add to $2$. Then she writes the two fractions whose numerator and denominator add to $3$, in increasing order of denominator. Then she writes the three fractions whose numerator and denominator sum to $4$ in increasing order of denominator. She continues in this way until she has written all the fractions whose numerator and denominator sum to at most $1000$. So Lizzie's list looks like: $$\frac11, \frac21, \frac12 , \frac31 , \frac22, \frac13, \frac41, \frac32, \frac23, \frac14, ..., \frac{1}{999}.$$ Let $p_k$ be the product of the first $k$ fractions in Lizzie's list. Find, with proof, the value of $p_1 + p_2 + ...+ p_{499500}$.

Given an acute-angled triangle $ABC$, in which $P$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $AC$, respectively. A point $H$ is taken inside the triangle and perpendiculars $HK$, $HS$, $HQ$ are lowered from it to the sides $AB$, $BC$, $AC$, respectively ($K \in AB$, $S \in BC$, $Q \in AC$). It turned out that $MK = MQ$, $NS = NK$, $PS=PQ$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$.

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.

The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function $f(x^3) -\frac{\sqrt3}{2} x^6$ also increases for all positive $x$.

Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.

Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.