Let $ABC$ be an equilateral triangle. Suppose $D$ is the point on $BC$ such that $BD+DC = 1:3$. Let the perpendicular bisector of $AD$ intersect $AB,AC$ at $E,F$ respectively. Prove that $49 \cdot [BDE] = 25 \cdot [CDF]$, where $[XYZ]$ denotes the area of the triangle $XYZ$.

Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?

In quadrilateral $ABCD$, $\angle B = \angle D = 90$ and $AC = BC + DC$. Point $P$ of ray $BD$ is such that $BP = AD$. Prove that line $CP$ is parallel to the bisector of angle $ABD$. [i](Proposed by A.Trigub)[/i]

A cube with sides 1m in length is filled with water, and has a tiny hole through which the water drains into a cylinder of radius $1\text{ m}$. If the water level in the cube is falling at a rate of $1 \text{ cm/s}$, at what rate is the water level in the cylinder rising?

How many ways are there to partition $7$ students into the groups of $2$ or $3$? $ \textbf{(A)}\ 70 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 210 \qquad\textbf{(D)}\ 280 \qquad\textbf{(E)}\ 630 $

Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. How many more bananas than apples did they sell? $\mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 4 \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 20$

Prove that a real number $x$ is rational if and only if the sequence $x, x+1, x+2, x+3, ..., x+n, ...$ contains, at least least three terms in geometric progression.

Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the $2018$th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors. A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Prove that for any natural number $n$ the following inequality holds $$4^n < (2n+1)C_{2n}^n$$

1. A finite sequence of integers $a_0,a_1,...,a_n$ is called quadratic if $|a_k -a_{k-1}| = k^2$ for $n\geq k\geq1$. (a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence with $a_0 = b$ and $a_n =c$. (b) Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_0 = 0$ and $a_n = 1997$

An $n$-sum of type $1$ is a finite sequence of positive integers $a_1,a_2,\ldots,a_r$, such that: $(1)$ $a_1+a_2+\ldots+a_r=n$; $(2)$ $a_1>a_2+a_3,a_2>a_3+a_4,\ldots, a_{r-2}>a_{r-1}+a_r$, and $a_{r-1}>a_r$. For example, there are five $7$-sums of type $1$, namely: $7$; $6,1$; $5,2$; $4,3$; $4,2,1$. An $n$-sum of type $2$ is a finite sequence of positive integers $b_1,b_2,\ldots,b_s$ such that: $(1)$ $b_1+b_2+\ldots+b_s=n$; $(2)$ $b_1\ge b_2\ge\ldots\ge b_s$; $(3)$ each $b_i$ is in the sequence $1,2,4,\ldots,g_j,\ldots$ defined by $g_1=1$, $g_2=2$, $g_j=g_{j-1}+g_{j-2}+1$; and $(4)$ if $b_1=g_k$, then $1,2,4,\ldots,g_k$ is a subsequence. For example, there are five $7$-sums of type $2$, namely: $4,2,1$; $2,2,2,1$; $2,2,1,1,1$; $2,1,1,1,1,1$; $1,1,1,1,1,1,1$. Prove that for $n\ge1$ the number of type $1$ and type $2$ $n$-sums is the same.

Worms grow at the rate of $1$ metre per hour. When they reach their maximal length of $1$ metre, they stop growing. A full-grown worm may be dissected into two not necessarily equal parts. Each new worm grows at the rate of $1$ metre per hour. Starting with $1$ full-grown worm, can one obtain $10$ full-grown worms in less than $1$ hour?

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals [asy] size(230); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(36), C=intersectionpoint(B--(9001,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(30)); label("$19$", (A+C)/2, dir(A--C)*dir(90)); label("$14$", (A+B)/2, dir(A--B)*dir(270)); [/asy] $\displaystyle \text{(A)} \ 2 \qquad \text{(B)} \ \frac{5}{2} \qquad \text{(C)} \ \frac{5}{2} - \sin \theta \qquad \text{(D)} \ \frac{5}{2} - \frac{1}{2} \sin \theta \qquad \text{(E)} \ \frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$

Five rays $\overrightarrow{OA}$,$\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$, and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$, $\angle COB = 2\angle BOA$, while $\angle DOC = 3\angle BOA$. If $E$, $O$, $A$ are collinear with $O$ between $A$ and $E$, what is the degree measure of $\angle DOB?$

If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$.

In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o$ . Prove that $AB + AC = 3BC$. (Matthew of Kursk)

Ivan has a parallelogram whose interior angles are $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ respectively, and all side lengths are integers. Is it possible that one of the diagonals has length $\sqrt{2024}$?

Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.

a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$. b) If the function $f: R \to R$ has the property: $$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram

Given a pyramid with square base $ ABCD $ and vertex $ S $. Find the shortest path whose starting and ending point is the point $ S $ and which passes through all the vertices of the base.

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.