Triangle $ABC$ is acute. Equilateral triangles $ABC',AB'C,A'BC$ are constructed externally to $ABC$. Let $BB'$ and $CC'$ intersect at $F$. Let $CC'$ intersect $AB$ at $C_1$ and $AA'$ intersect $BC$ at $A_1$, and let $A_1C_1$ intersect $AC$ at $D$. If $A'F=23$, $CF=13$, and $DF=24$, find $BD$. [i]2017 CCA Math Bonanza Team Round #10[/i]

Prove that the equation $x^2 + x + 1 = py$ has solution $(x,y)$ for the infinite number of simple $p$.

Let $T$ be an isosceles right triangle. Let $S$ be the circle such that the difference in the areas of $T \cup S$ and $T \cap S$ is the minimal. Prove that the centre of $S$ divides the altitude drawn on the hypotenuse of $T$ in the golden ratio (i.e., $\frac{(1 + \sqrt{5})}{2}$)

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

Given $a,b,c$ ,$(a, b,c \in (0,2)$), with $a + b + c = ab+bc+ca$, prove that $$\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{c^2}{c^2-c+1} \le 3$$ (D. Pirshtuk)

Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

Let $a, b$ and $c$ be real numbers such that $$\lceil a \rceil + \lceil b \rceil + \lceil c \rceil + \lfloor a + b \rfloor + \lfloor b + c \rfloor + \lfloor c + a \rfloor = 2020$$ Prove that $$\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor + \lceil a + b + c \rceil \ge 1346$$ Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. That is, $\lfloor x \rfloor$ is the unique integer satisfying $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$, and $\lceil x \rceil$ is the unique integer satisfying $\lceil x \rceil - 1 < x \le \lceil x \rceil$. [i]Proposed by Ariel García[/i]

Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.

What percentage of twenty thousand is a quarter of a million?

Let $ABC$ be an acute angled triangle. Let $B' , A'$ be points on the perpendicular bisectors of $AC, BC$ respectively such that $B'A \perp AB$ and $A'B \perp AB$. Let $P$ be a point on the segment $AB$ and $O$ the circumcenter of the triangle $ABC$. Let $D, E$ be points on $BC, AC$ respectively such that $DP \perp BO$ and $EP \perp AO$. Let $O'$ be the circumcenter of the triangle $CDE$. Prove that $B', A'$ and $O'$ are collinear. [i]Steve Dinh[/i]

Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear. Tran Quang Hung, High School of Natural Sciences, Hanoi National University

A girls' camp is located $ 300$ rods from a straight road. On this road, a boys' camp is located $ 500$ rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is: $ \textbf{(A)}\ 400\text{ rods} \qquad\textbf{(B)}\ 250\text{ rods} \qquad\textbf{(C)}\ 87.5\text{ rods} \qquad\textbf{(D)}\ 200\text{ rods}\\ \textbf{(E)}\ \text{none of these}$

Let $ABC$ be an equilateral triangle of side length $1.$ For a real number $0<x<0.5,$ let $A_1$ and $A_2$ be the points on side $BC$ such that $A_1B=A_2C=x,$ and let $T_A=\triangle AA_1A_2.$ Construct triangles $T_B=\triangle BB_1B_2$ and $T_C=\triangle CC_1C_2$ similarly. There exist positive rational numbers $b,c$ such that the region of points inside all three triangles $T_A,T_B,T_C$ is a hexagon with area $$\dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}.$$ Find $(b,c).$

Let $\frac{2}{\sqrt5+1}\leq p < 1$, and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$'s and $\pm 1$'s for which $\sum_{n=1}^\infty e_np^n=0$, we also have $\sum_{n=1}^\infty e_na_n=0$. Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$. [Z. Daroczy, I. Katai]

For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$ Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

In a soccer league with $2020$ teams every two team have played exactly once and no game have lead to a draw. The participating teams are ordered first by their points (3 points for a win, 1 point for a draw, 0 points for a loss) then by their goal difference (goals scored minus goals against) in a normal soccer table. Is it possible for the goal difference in such table to be strictly increasing from the top to the bottom? [i]Proposed by Abolfazl Asadi[/i]

Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ . .

First $a$ is chosen at random from the set $\{1,2,3,\ldots,99,100 \}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is $\text{(A)} \ \frac1{16} \qquad \text{(B)} \ \frac18 \qquad \text{(C)} \ \frac{3}{16}\qquad \text{(D)} \ \frac15 \qquad \text{(E)} \ \frac14$

Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.

Let $a_1,a_2,...,a_{2007}$ be real number such that $ a_1+a_2+...+a_{2007}\geq 2007^{2}$ and $a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1 $. Prove that $ a_k\in[2006;2008]$ for all $k\in\left \{ 1,2,...,2007 \right \}$

A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$ with the circumcircles of triangles $CDM_1$ and $ABM_2$. Prove that $X_1X_2 // Y_1Y_2$.

Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.