In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.

Two points $J$ and $H$ lie $26$ units apart on a given plane. Let $M$ be the locus of points $T$ on this plane such that $JT^2 + HT^2 = 2020$. Then, M encloses a region on the plane with area $a$ and perimeter $p$. If $q$ and $r$ are coprime positive integers and $\frac{a}{p} = \frac{q}{r}$ , then compute $q + r$.

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \quad \text{for all } n \geq 1. \end{align*} Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$. [i](South Africa)[/i]

Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?

In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.

A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$. [asy] unitsize (3 cm); pair A, B, C, D, E, F; D = (0,0); A = dir(250); B = dir(290); C = B + D - A; E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D); F = extension(A, E, B, C); draw(A--B--C--D--cycle); draw(A--F--D--B); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, S); dot("$F$", F, SE); [/asy]

Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle); [/asy] $ \textbf{(A)}\ 4.1 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.3 \qquad \textbf{(D)}\ 4.4 \qquad \textbf{(E)}\ 4.5$

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a\cdot 21^b}$ is a positive integer. [i]Proposed by Aidan Duncan[/i]

$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. [i]N. Agakhanov, I. Bogdanov [/i]

Consider the sequence $(x_n)_{n\in\mathbb{N^*}}$ such that $$x_0=0,\quad x_1=2024,\quad x_n=x_{n-1}+x_{n-2}, \forall n\geq2.$$ Prove that there is an infinity of terms in this sequence that end with $2024.$

The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$. Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.

Find a positive integer $N$ and $a_1, a_2, \cdots, a_N$ where $a_k = 1$ or $a_k = -1$, for each $k=1,2,\cdots,N,$ such that $$a_1 \cdot 1^3 + a_2 \cdot 2^3 + a_3 \cdot 3^3 \cdots + a_N \cdot N^3 = 20162016$$ or show that this is impossible.

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

Let regular tetrahedron $ABCD$ have center $O$. Find $\tan^2(\angle AOB)$. [i]2022 CCA Math Bonanza Individual Round #6[/i]

Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

6) In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 g$, $M_2 = 200 g$, and $M_4 = 500 g$. What is the value of $M_3$ for the system to be in static equilibrium? A) 300 g B) 400 g C) 500 g D) 600 g E) 700 g

An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.