Found problems: 85335
Let $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$. If $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$ for distinct numbers $a,b,c$, what are all possible values of $a+b+c$?
[i]2019 CCA Math Bonanza Individual Round #12[/i]
Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$
Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$
[i]D. Zmiaikou[/i]
Find all prime numbers $p$, for which the number $p + 1$ is equal to the product of all the prime numbers which are smaller than $p$.
A circle is divided by $2018$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?
Let $ f : [0, 1] \to\ R $ be a function such that :-
$1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ .
$2.)$ $f(1) = 1$ .
$3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ .
Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.
Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.
Find all triples $ (a,b,c)$ of integers with $ abc\equal{}1989$ and $ a\plus{}b\minus{}c\equal{}89$.
Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$?
$ \textbf{(A)}\ 45 \qquad
\textbf{(B)}\ 63 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 201 \qquad
\textbf{(E)}\ 1005$
From digits $0$, $1$, $3$, $4$, $7$ and $9$ were written $5$ digit numbers which all digits are different. How many numbers from them are divisible with $5$
The following light grid is given:
\begin{tabular}{cccc}
o & o & o & o \\
o & o & o & o \\
o & o & o & o \\
o & o & o & o
\end{tabular}
where `o` represents a switched-off light and `•` represents a switched-on light. Each time a light is pressed, it toggles its state (on/off) as well as the state of its four adjacent neighbors (left, right, above, below). The bottom edge lights are considered to be immediately above the top edge lights, and the same applies to the lateral edges.The right figure illustrates the effect of pressing a light in a corner.
Pressing a certain combination of lights results in all lights turning on. Prove that all lights must have been pressed at least once.
A triangle with integral sides has perimeter $8$. The area of the triangle is
$\textbf{(A) }2\sqrt{2}\qquad
\textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad
\textbf{(C) }2\sqrt{3}\qquad
\textbf{(D) }4\qquad
\textbf{(E) }4\sqrt{2}$
Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy:
(a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$.
(b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$.
Find the general term of $\{ a_{n}\}$.
Let $ABC$ be an acute triangle. Let $D, E,$ and $F$ be the feet of altitudes from $A, B,$ and $C$ to sides $BC, CA,$ and $AB$, respectively, and let $Q$ be the foot of altitude from A to line $EF$ . Given that $AQ = 20, BC = 15,$ and $AD = 24$, compute the perimeter of triangle $DEF.$
Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.
$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of
$$ \sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|$$
The point $O{}$ is given in the plane. Find all natural numbers $n{}$ for which $n{}$ points in the plane can be colored red, so that for any two red points $A{}$ and $B{}$ there is a third red point $C{}$ is such that $O{}$ lies strictly inside the triangle $ABC$.
[i]From the folklore[/i]
Let $\sigma(n)$ be the sum of all the positive divisors of $n$. Let $a$ be the smallest positive integer greater than or equal to $2015$ for which there exists some positive integer $n$ satisfying $\sigma(n)=a$. Finally, let $b$ be the largest such value of $n$. Compute $a+b$.
Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$.
[i]Ciprus[/i]
Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.
a) Prove that the lines $AC$ and $DE$ are parallel
b) Prove that $AE=CD$
The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.
Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$
$\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85$
Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that
$$
\sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.
Blake and Jenny each took four $100$ point tests. Blake averaged $78$ on the four tests. Jenny scored $10$ points higher than Blake on the first test, $10$ points lower on the second test, and $20$ points higher on both the third and fourth test.
What is the difference between Blake's average on the four tests and Jenny's average on the four tests?
$\textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 25 \qquad
\textbf{(E)}\ 40$