Found problems: 85335
For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with
\[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a $12 \times 12$ board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.
Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?
Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.
We have four white equilateral triangles of $3$ cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?
Let $ ABC$ be a triangle, all of whose angles are greater than $ 45^{\circ}$ and smaller than $ 90^{\circ}$.
(a) Prove that one can fit three squares inside $ ABC$ in such a way that: (i) the three squares are equal (ii) the three squares have common vertex $ K$ inside the triangle (iii) any two squares have no common point but $ K$ (iv) each square has two opposite vertices onthe boundary of $ ABC$, while all the other points of the square are inside $ ABC$.
(b) Let $ P$ be the center of the square which has $ AB$ as a side and is outside $ ABC$. Let $ r_{C}$ be the line symmetric to $ CK$ with respect to the bisector of $ \angle BCA$. Prove that $ P$ lies on $ r_{C}$.
$S$ is natural, and $S=d_{1}>d_2>...>d_{1000000}=1$ are all divisors of $S$. What minimal number of divisors can have $d_{250}$?
If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest?
$ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ \text{no one is always largest} $
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $ A$, $ B$, and $ C$ start with $ 15$, $ 14$, and $ 13$ tokens, respectively. How many rounds will there be in the game?
$ \textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 37 \qquad
\textbf{(C)}\ 38 \qquad
\textbf{(D)}\ 39 \qquad
\textbf{(E)}\ 40$
Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
$ \textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39$
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$
The CCA Math Banana$^{\text{TM}}$ costs \$$100$. The cost rises $10$% then drops $10$%. Now what is the cost of the CCA Math Banana$^{\text{TM}}$?
[i]2018 CCA Math Bonanza Lightning Round #1.2[/i]
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is
[asy]
for (int a=0; a <= 3; ++a)
{
for (int b=0; b <= 3-a; ++b)
{
fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey);
}
}
for (int c=0; c <= 3; ++c)
{
draw((c,0)--(c,4-c),linewidth(1));
draw((0,c)--(4-c,c),linewidth(1));
draw((c+1,0)--(0,c+1),linewidth(1));
}
label("$8$",(2,0),S);
label("$8$",(0,2),W);
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64$
Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.
Solve the following equation where $x$ is a real number: $\lfloor x^2 \rfloor -10\lfloor x \rfloor + 24 = 0$
For positive numbers $ x_1, x_2, \ldots, x_n $ $ (n \geq 1) $ the following equality holds $$ \frac {1} {{1 + x_1}} + \frac {1} {{1 + x_2}} + \ldots + \frac {1} {{1 + x_n}} = 1. $$ Prove that $ x_1 \cdot x_2 \cdot \ldots \cdot x_n \geq (n-1) ^ n. $
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$.
[i]M. Sonkin[/i]
Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.
A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square).
(A. Perlin)
Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $