How many consequtive numbers are there in the set of positive integers in which powers of all prime factors in their prime factorizations are odd numbers? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 15 $

Let $X=\left\{1,2,3,4,5,6\right\}$. How many non-empty subsets of $X$ do not contain two consecutive integers? $\text{(A) }16\qquad\text{(B) }18\qquad\text{(C) }20\qquad\text{(D) }21\qquad\text{(E) }24$

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

[b]a)[/b] Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$. [b]b)[/b] Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$. [b]c)[/b] Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$. [i]Proposed by Mostafa Eynollahzade[/i]

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

How many axes of symmetry can a heptagon have?

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the following assertions: a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$. b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.

Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\Gamma$ be the branch $x> 0$ of the hyperbola $x^2 - y^2 = 1.$ Let $P_0, P_1,..., P_n$ different points of $\Gamma$ with $P_0 = (1, 0)$ and $P_1 = (13/12, 5/12)$. Let $t_i$ be the tangent line to $\Gamma$ at $P_i$. Suppose that for all $i \geq 0$ the area of ​​the region bounded by $t_i, t_{i +1}$ and $\Gamma$ is a constant independent of $i$. Find the coordinates of the points $P_i$.

Non-degenerate triangle $ABC$ has $AB=20$, $AC=17$, and $BC=n$, an integer. How many possible values of $n$ are there? [i]2017 CCA Math Bonanza Lightning Round #2.2[/i]

Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$

Triangle $ABC$ is given, where $AC<BC$ and $\angle ACB=60^\circ\!\!.$ Point $D$, distinct from $A$, lies on the segment $AC$ such that $AB=BD$, and point $E$, distinct from $B$, lies on the line $BC$ such that $AB=AE$. Prove that $\angle DEC=30^\circ$.

Given $I_0 = \{-1,1\}$, define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$, where $y$ ranges over all elements of $I_{n-1}$. Determine the union of the sets $I_n$ over all nonnegative integers $n$.

Let $p = 2017$. If $A$ is an $n\times n$ matrix composed of residues $\pmod{p}$ such that $\det A\not\equiv 0\pmod{p}$ then let $\text{ord}(A)$ be the minimum integer $d > 0$ such that $A^d\equiv I\pmod{p}$, where $I$ is the $n\times n$ identity matrix. Let the maximum such order be $a_n$ for every positive integer $n$. Compute the sum of the digits when $\sum_{k = 1}^{p + 1} a_k$ is expressed in base $p$. [i]Proposed by Ashwin Sah[/i]

Four masses $m$ are arranged at the vertices of a tetrahedron of side length $a$. What is the gravitational potential energy of this arrangement? (A) $-2\frac{Gm^2}{a}$ (B) $-3\frac{Gm^2}{a}$ (C) $-4\frac{Gm^2}{a}$ (D) $-6\frac{Gm^2}{a}$ (E) $-12\frac{Gm^2}{a}$

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

In a country, there are \( n \) cities. Each pair of cities is connected either by a paved road or a dirt road. It is known that there exists a pair of cities such that it is impossible to travel between them using only paved roads. Show that, in this case, it is possible to travel between any two cities using only dirt roads.

Circle $\Omega$ with center $O$ is the circumcircle of an acute triangle $\triangle ABC$ with $AB<BC$ and orthocenter $H$. On the line $BO$ there is point $D$ such that $O$ is between $B$ and $D$ and $\angle ADC= \angle ABC$ . The semi-line starting at $H$ and parallel to $BO$ wich intersects segment $AC$ , intersects $\Omega$ at $E$. Prove that $BH=DE$.

Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]

The positive numbers $a, b$ and $c$ satisfy $a \ge b \ge c$ and $a + b + c \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 \le 1$ . (F . L . Nazarov)

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \[\sum_{i=1}^{1000} g(i).\] [i]Proposed by Nathan Ramesh

Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$ where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$. Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.

On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?