Found problems: 85335
Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$.
[i] (Vlad Robu) [/i]
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.
For all reals $a, b, c,d $ prove the following inequality:
$$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$
Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball. It is assumed that the balls can only touch externally.
Let $n{}$ and $k{}$ be natural numbers, with $n > 2k$. In the deck of cards, each card contains a subset of the set $\{1, 2, \ldots , n\}$ consisting of at least $k+1$, but no more than $n-k$ elements. Each $m$-element set is written exactly on $m-k$ cards. Is it possible to split these cards into $n- 2k$ stacks so that in each stack all subsets on the cards are different, and any two of them intersect?
Teams $\text{A}$ and $\text{B}$ are playing a series of games. If the odds for either to win any game are even and Team $\text{A}$ must win two or Team $\text{B}$ three games to win the series, then the odds favoring Team $\text{A}$ to win the series are
$\textbf{(A) }11\text{ to }5\qquad\textbf{(B) }5\text{ to }2\qquad\textbf{(C) }8\text{ to }3\qquad\textbf{(D) }3\text{ to }2\qquad \textbf{(E) }13\text{ to }6$
An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$.
[img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]
Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.
Consider the function
\[
f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots},
\]
where $n$ is a positive integer. Express $f(x,n+1)$ rationally in terms of $f(x,n)$ and $x$. Hence, or otherwise, evaluate $\textstyle\lim_{n\to\infty}f(x,n)$ for suitable fixed values of $x$. (The symbols $\textstyle\binom nr$ represent the binomial coefficients.)
Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.
For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He de nes the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly $90$ oat pieces and $9$ marshmallow pieces, and exclaims, "This is such an unlucky bowl!" How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?
Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$
Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the
$$A\circ B=\sqrt A B\sqrt B$$
operation. Prove that for a given pair $A, B$ of positive operators the identity
$$(A\circ B)\circ C=A\circ (B\circ C)$$
holds for all positive operator $C$ if and only if $AB=BA$.
Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{33}{7} \qquad
\textbf{(C)}\ \frac{39}{7} \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ \frac{99}{10} $
A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle $I$ to that of circle $II$, then $r$ equals:
$\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3$
A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.
Depending on the real number \( a \), find all polynomials \( P(x) \) with real coefficients such that
\[
(x^3 - ax^2 + 1)P(x) = (x^3 + ax^2 + 1)P(x-1)
\]
for every real number \( x \).
Let $a_1,a_2,...,a_n$ be pairwise distinct real numbers and $m$ be the number of distinct sums $a_i +a_j$ (where $i \ne j$). Find the least possible value of $m$.
For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.
Burrito Bear has a white unit square. She inscribes a circle inside of the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
[i]2022 CCA Math Bonanza Individual Round #4[/i]
On square $ABCD$, point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC}$, so that $BE=EF=FD=30$. Find the area of square $ABCD$.
The triangle $ABC$ is right-angled at $A$. Its incentre is $I$, and $H$ is the foot of the perpendicular from $I$ on $AB$. The perpendicular from $H$ on $BC$ meets $BC$ at $E$, and it meets the bisector of $\angle ABC$ at $D$. The perpendicular from $A$ on $BC$ meets $BC$ at $F$. Prove that $\angle EFD = 45^o$
Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.
It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?