A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. [i]Original formulation:[/i] A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

Let $P(n)$ denote the product of digits of $n$. Find the number of positive integers $n \leq 2024$ where $P(n)$ is divisible by $n$.

Denote $H$ and $I$ as the orthocenter and incenter, respectively, of triangle $\triangle ABC$. Let $M$ be the midpoint of $\overline{BC}$. Prove that $\angle{HIM} = 90^\circ$ if and only if $AB + AC = 2BC$. [i]Eric Shen and Howard Halim[/i]

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]

Let $p$ - a prime, where $p> 11$. Prove that there exists a number $k$ such that the product $p \cdot k$ can be written in the decimal system with only ones.

Which digit of $0.12345$, when changed to $9$, gives the largest number? $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 $

Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$, determine the largest possible value of $m$.

If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy $lcm (a, b, c, d) < 1000$ and $a+b = c+d$, compute the largest possible value of $a + b$.

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .

For all positive real numbers $a_1, a_2, \dots, a_n$, prove that $$ \frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot \frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot \frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot \frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.$$

Let $M=\{1,2,...,n\}$. Prove that the number of pairs $(A,a)$, where $A\subset M$ and $a$ is a permutation of $M$, for which $a(A)\cap A=\emptyset $, is equal to $n!.F_{n+1}$, where $F_{n+1}$ is the $n+1$ member of the Fibonacci sequence.

A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines? $\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$

Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.

Say that an ordered triple $(a, b, c)$ is [i]pleasing[/i] if (a) $a$, $b$, and $c$ are in the set $\{ 1, 2, \dots, 17 \}$, and (b) both $b - a$ and $c - b$ are greater than $3$, and at least one of them is equal to $4$. How many pleasing triples are there?

Call an [i]ordered[/i] triple $(a,b,c)$ [i]$d$-tall[/i] if there exists a triangle with side lengths $a,b,c$ and the height to the side with length $a$ is $d$. Suppose that for some positive integer $k$, there are exactly $210$ $k$-tall ordered triples of positive integers. How many $k$-tall ordered triples $(a,b,c)$ are there such that a triangle $ABC$ with $BC=a,CA=b,AB=c$ satisfies both $\angle{B}<90^\circ$ and $\angle{C}<90^\circ$? [i]2020 CCA Math Bonanza Team Round #8[/i]

Let $ABC$ be a triangle, and let $D$ be the foot of the $A-$altitude. Points $P, Q$ are chosen on $BC$ such that $DP = DQ = DA$. Suppose $AP$ and $AQ$ intersect the circumcircle of $ABC$ again at $X$ and $Y$. Prove that the perpendicular bisectors of the lines $PX$, $QY$, and $BC$ are concurrent. [i]Proposed by Pranjal Srivastava[/i]

Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$

Let $ n$ denote the smallest positive integer that is divisible by both $ 4$ and $ 9$, and whose base-$ 10$ representation consists of only $ 4$'s and $ 9$'s, with at least one of each. What are the last four digits of $ n$? $ \textbf{(A)}\ 4444\qquad \textbf{(B)}\ 4494\qquad \textbf{(C)}\ 4944\qquad \textbf{(D)}\ 9444\qquad \textbf{(E)}\ 9944$

Find all positive integers $n$ such that the elements of $$\{1,2,...,2n+1\}-\{n+1\}$$ can be partitioned into two groups with the same number of elements and the same sum of their elements.

Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

[u]Version for Nordic Countries[/u] Six pine trees grow on the shore of a circular lake. It is known that a treasure is submerged at the mid-point $T$ between the intersection points of the altitudes of two triangles, the vertices of one being at three of the $6$ pines, and the vertices of the second one at the other three pines. At how many points $T$ must one dive to find the treasure? [u]Version for Tropical Countries[/u] A captain finds his way to Treasure Island, which is circular in shape. He knows that there is treasure buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$ and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. He finds the trees all right, but does not know which tree is denoted by which letter. What is the maximum number of points at which the captain has to dig in order to recover the treasure? (S Markelov)

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$