Let $ABCD$ be a convex quadrilateral. The common external tangents to circles $(ABC)$ and $(ACD)$ meet at point $E$, the common external tangents to circles $(ABD)$ and $(BCD)$ meet at point $F$. Let $F$ lie on $AC$, prove that $E$ lies on $BD$.

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

Alice thinks about a natural number in her mind. Bob tries to find that number by asking him the following 10 questions: [list] [*]Is it divisible by 1? [*]Is it divisible by 2? [*]Is it divisible by 3? [*]... [*]Is it divisible by 9? [*]Is it divisible by 10? [/list] Alice's answer to all questions except one was "yes". When she answers "no", she adds that "the greatest common factor of the number I have in mind and the divisor in the question you asked is 1”. According to this information, to which question did Alice answer "no"?

Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$. [i](tatari/nightmare)[/i]

A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$, and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$. Find $a_{100}$.

\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited

Let $ABC$ be an acute triangle. Construct a point $X$ on the different side of $C$ with respect to the line $AB$ and construct a point $Y$ on the different side of $B$ with respect to the line $AC$ such that $BX=AC$, $CY=AB$, and $AX=AY$. Let $A'$ be the reflection of $A$ across the perpendicular bisector of $BC$. Suppose that $X$ and $Y$ lie on different sides of the line $AA'$, prove that points $A$, $A'$, $X$, and $Y$ lie on a circle.

Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$

Suppose there are $n$ points on the plane, no three of which are collinear. Draw $n-1$ non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a [i]network[/i]. Given a network, we may assign labels from $1$ to $n-1$ to each segment such that each segment gets a different label. Define a [i]spin[/i] as the following operation: $\bullet$ Choose a point $v$ and rotate the labels of its adjacent segments clockwise. Formally, let $e_1,e_2,\cdots,e_k$ be the segments which contain $v$ as an endpoint, sorted in clockwise order (it does not matter which segment we choose as $e_1$). Then, the label of $e_{i+1}$ is replaced with the label of $e_{i}$ simultaneously for all $1 \le i \le k$. (where $e_{k+1}=e_{1}$) A network is [i]nontrivial[/i] if there exists at least $2$ points with at least $2$ adjacent segments each. A network is [i]versatile[/i] if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers $n \ge 5$ such that any nontrivial network with $n$ points is versatile. [i]Proposed by Yeoh Zi Song[/i]

Find the largest possible value of $a$ such that there exist real numbers $b,c>1$ such that \[a^{\log_b c}\cdot b^{\log_c a}=2023.\] [i]Proposed by Howard Halim[/i]

Source: 2017 Canadian Open Math Challenge, Problem B4 ----- Numbers $a$, $b$ and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.

We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes. Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$

Which of the following is closest to $\sqrt{65}-\sqrt{63}$? $ \textbf{(A)}\ .12 \qquad\textbf{(B)}\ .13 \qquad\textbf{(C)}\ .14 \qquad\textbf{(D)}\ .15 \qquad\textbf{(E)}\ .16 $

You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year $ 2003$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

There are 20 blue points on the circle and some red inside so no three are collinear. It turned out that there exists $1123$ triangles with blue vertices having 10 red points inside. Prove that all triangles have 10 red points inside

For each positive integer $n$, let $rad(n)$ denote the product of the distinct prime factors of $n$. Show that there exists integers $a,b > 1$ such that $gcd(a,b)=1$ and $$rad(ab(a+b)) < \frac{a+b}{2024^{2024}}$$. For example, $rad(20)=rad(2^2\cdot 5)=2\cdot 5=10$.

Compute all real solutions $(x,y)$ with $x\geq y$ that satisfy the pair of equations \begin{align*} xy&=5\\ \frac{x^2+y^2}{x+y}&=3. \end{align*}

The following sets of points are considered in the plane: $A=\{$ affixes of complexes $z$ such that arg $(z - (2 + 3i))=\pi /4\}$, $B =\{$ affixes of complexes $z$ such that mod $( z- (2 + i)<2\}$. Determine the orthogonal projection on the $X$ axis of $A \cap B$.

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$ a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$ b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.

We say that a positive integer $n$ is [i]divable [/i] if there exist positive integers $1 < a < b < n$ such that, if the base-$a$ representation of $n$ is $\sum_{i=0}^{k_1} a_ia^i$ , and the base-$b$ representation of $n$ is $\sum_{i=0}^{k_2} b_ib^i$ , then for all positive integers $c > b$, we have that $\sum_{i=0}^{k_2} b_ic^i$ divides $\sum_{i=0}^{k_1} a_ic^i$. Find the number of non-divable $n$ such that $1 \le n \le 100$.

A $3\times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules: (1) Both the first three columns and the last three columns contain the integers 1 through 9. (2) No numbers appear more than once in a given row. Let $N$ be the number of ways to fill the grid and let $k$ be the largest positive integer such that $2^k$ divides $N$. What is k?

Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container? $\textbf{(A) } \frac{5}{8} \qquad \textbf{(B) } \frac{4}{5} \qquad \textbf{(C) } \frac{7}{8} \qquad \textbf{(D) } \frac{9}{10} \qquad \textbf{(E) } \frac{11}{12}$