Consider the binomial coefficients $\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)$. Determine all positive integers $n$ for which $\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.

If $a,b,c$ are positive three real numbers such that $| a-b | \geq c , | b-c | \geq a, | c-a | \geq b$ . Prove that one of $a,b,c$ is equal to the sum of the other two.

Let $a$ , $b$ , $c$ be positive real numbers such that $a+b+c=a^2+b^2+c^2$ . Prove that : \[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]

Let $\omega$ be a circle. Points $A,B,C,X,D,Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB,BC$, respectively. Prove that points $B,E,F,Y$ lie on a single circle.

Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$ for all $x,y\in Q^+ .$

Show that there exists a positive integer $n$ such that the decimal representations of $3^n$ and $7^n$ both start with the digits $10$.

Are there such natural $n$, that exist polynomial of degree $n$ and with $n$ different real roots, and a) $P(x)P(x+1)=P(x^2)$ b) $P(x)P(x+1)=P(x^2+1)$

Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011} dx$.

Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$

Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative.

Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise. Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class. [i]David Yang.[/i]

Given that $\sec x+\tan x=2018$, compute $\csc x+\cot x$.

Mark's teacher is randomly pairing his class of $16$ students into groups of $2$ for a project. What is the probability that Mark is paired up with his best friend, Mike? (There is only one Mike in the class) [i]2015 CCA Math Bonanza Individual Round #3[/i]

Consider $0<a<T$, $D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}$, and let $f:D\to \mathbb{R}$ a $T-$periodic and differentiable function which satisfies $f' > 1$ on $(0, a)$ and $$f(0)=0,\lim_{\substack{x\to a\\x<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a\\ x<a}}\frac{f'(x)}{f^2(x)}=1.$$ [list] [*]Prove that for every $n\in \mathbb{N}^*$, the equation $f(x)=x$ has a unique solution in the interval $(nT, nT+a)$ , denoted $x_n$.[/*] [*]Let $y_n=nT+a-x_n$ and $z_n=\int_0^{y_n}f(x)\text{d}x$. Prove that $\lim_{n\to \infty}{y_n}=0$ and study the convergence of the series $\sum_{n=1}^{\infty}{y_n}$ and $\sum_{n=1}^{n}{z_n}$. [/list]

Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).

Find the number of ways to partition a set of $10$ elements, $S = \{1, 2, 3, . . . , 10\}$ into two parts; that is, the number of unordered pairs $\{P, Q\}$ such that $P \cup Q = S$ and $P \cap Q = \emptyset$.

On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?

[u]Set 1[/u] [b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs? [b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$? [b]G3.[/b] Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had. [u]Set 2[/u] [b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? [b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$. [b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$. [u]Set 3[/u] [b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have? [b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$. [b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$ and let $H$ be the foot of the altitude from $B$. Let $P$ and $Q$ be orthogonal projections of $A$ and $C$ on the bisector of the angle $B$. Prove that the four points $H,P,M$ and $Q$ lie on the same circle.

The geometric series $ a \plus{} ar \plus{} ar^{2} \plus{} ...$ has a sum of $ 7$, and the terms involving odd powers of $ r$ have a sum of $ 3$. What is $ a \plus{} r$? $ \textbf{(A)}\ \frac {4}{3}\qquad \textbf{(B)}\ \frac {12}{7}\qquad \textbf{(C)}\ \frac {3}{2}\qquad \textbf{(D)}\ \frac {7}{3}\qquad \textbf{(E)}\ \frac {5}{2}$

The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.

Let $C$ be the set of non-negative integers which can be expressed as $1999s+2000t$, where $s$ and $t$ are also non-negative integers. (a) Show that $3,994,001$ is not in $C$. (b) Show that if $0\leq n \leq 3,994,001$ and $n$ is an integer not in $C$, then $3,994,001-n$ is in $C$.

Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$

In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.