Found problems: 85335
You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.
$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition [i]honest[/i], if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
$\text{(A)}\ 30 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 70$
You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$.
Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are
integers
$\frac{p^2 + 2q}{q+r}, \frac{q^2+9r}{r+p}, \frac{r^2+3p}{p+q}$
[i]Proposed by Tajikistan[/i]
In an acute triangle $ABC$ the heights $AD, BE$ and $CF$ intersecting at $H{}$. Let $O{}$ be the circumcenter of the triangle $ABC$. The tangents to the circle $(ABC)$ drawn at $B{}$ and $C{}$ intersect at $T{}$. Let $K{}$ and $L{}$ be symmetric to $O{}$ with respect to $AB$ and $AC$ respectively. The circles $(DFK)$ and $(DEL)$ intersect at a point $P{}$ different from $D{}$. Prove that $P, D$ and $T{}$ lie on the same line.
[i]Proposed by Don Luu (Vietnam)[/i]
A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards.
Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$
$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.
Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that
$$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$
Determine the minimum value of $x_1+x_2+x_3+...+x_n$.
[i]Proposed by Loh Kwong Weng[/i]
A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so.
Determine the largest possible value of $m$.
There are $86400$ seconds in a day, which can be deduced from the conversions between seconds, minutes, hours, and days. However, the leading scientists decide that we should decide on $3$ new integers $x, y$, and $z$, such that there are $x$ seconds in a minute, $y$ minutes in an hour, and $z$ hours in a day, such that $xyz = 86400$ as before, but such that the sum $x + y + z$ is minimized. What is the smallest possible value of that sum?
(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression.
(b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$.
[i]2020 CCA Math Bonanza Individual Round #13[/i]
Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i$ and $j$ are distinct.
[i]Miroslav Marinov, Bulgaria[/i]
Let $\lVert u\rVert$ denote the distance from the real number $u$ to the nearest integer. For positive integers $n$, let
$$a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.$$Determine $\lim_{n\to\infty}a_n$.
Some coins are placed on a $20 \times 13$ board. Two coins are called [i]neighbours[/i] if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?
Given a triangle $ABC$ with $\angle C = 90^o$.
a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle.
b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
Let $p(x)=x^4-4x^3+2x^2+ax+b$. Suppose that for every root $\lambda$ of $p$, $\frac{1}{\lambda}$ is also a root of $p$. Then $a+b=$
[list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list]
Find an integral formula for the solution of the differential equation
$$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$
for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$
We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$
Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.