$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

Prove, that for any positive real numbers $a, b, c$ who satisfy $a^2+b^2+c^2=1$ the following inequality holds. $\sqrt{\frac{1}{a}-a}+\sqrt{\frac{1}{b}-b}+\sqrt{\frac{1}{c}-c} \geq \sqrt{2a}+\sqrt{2b}+\sqrt{2c}$

Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\tfrac{AW}{BW} = \tfrac29$ , $\tfrac{AX}{CX} = \tfrac56$ , and $\tfrac{DY}{EY} = \tfrac92$. The ratio $\tfrac{EZ}{FZ}$ can then be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.6)); real r = 3/11, s = 0.52, l = 33, d=5.5; pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0); pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F); draw(E--D--F--B--A--C^^W--Z); dot("$A$",A,N); dot("$B$",B,S); dot("$C$",C,S); dot("$D$",D,N); dot("$E$",E,S); dot("$F$",F,S); dot("$W$",W,0.6*NW); dot("$X$",X,0.8*NE); dot("$Y$",Y,dir(100)); dot("$Z$",Z,dir(70)); [/asy]

Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.

In $\Delta ABC$, $\angle ABC=120^\circ$. The internal bisector of $\angle B$ meets $AC$ at $D$. If $BD=1$, find the smallest possible value of $4BC+AB$.

Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.

The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square. [DIAGRAM NEEDED]

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

Given $\triangle ABC$, whose all sides have different length. Point $P$ is chosen on altitude $AD$. Lines $BP$ and $CP$ intersect lines $AC, AB$ respectively and point $X, Y$.It is given that $AX=AY$. Prove that there is circle, whose centre lies on $BC$ and is tangent to sides $AC$ and $AB$ at points $X,Y$.

Let $n$ be a fixed positive integer. Find the minimum value of $$\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}$$ where $x_1,x_2,\dots,x_n$ are distinct positive integers.

Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$. Points $A$ and $P$ lie on the same side of line $BC$. Point $R$ is an orthogonal projection of point $P$ on line $AC$. Point $S$ is middle of line segment $AQ$. Show that points $A, B, R, S$ lie on one circle.

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by: \[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \] Show that there exist a real number $a > 0$ such that: [b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$ [b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$

Natural numbers $a$ and $b$ are given such that the number $$ P = \frac{[a, b]}{a + 1} + \frac{[a, b]}{b + 1} $$ Is a prime. Prove that $4P + 5$ is the square of a natural number.

Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$, $AC$ and $BD$, $AF$ and $DE$, $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.

A confused cockroach is initially at vertex $A$ of a cube $ABCDEFGH$ with edges measuring $1$ meter, as shown in the figure. Every second, the cockroach moves $1$ meter, always choosing to go to one of the three adjacent vertices to its current position. For example, after $1$ second, the cockroach could stop at vertex $B$, $D$, or $E$. (a) In how many ways can the cockroach stop at vertex $G$ after $3$ seconds? (b) Is it possible for the cockroach to stop at vertex A after exactly $2023$ seconds? (c) In how many ways can the cockroach stop at A after exactly $2024$ seconds? Note: One way for the cockroach to stop at a vertex after a certain number of seconds differs from another way if, at some point, the cockroach is at different vertices in the trajectory. For example, there are $2$ ways for the cockroach to stop at $C$ after $2$ seconds: one of them passes through $A$, $B$, $C$, and the other through $A$, $D$, $C$. [img]https://cdn.discordapp.com/attachments/954427908359876608/1299721377124847616/Screenshot_2024-10-16_173123.png?ex=671e3b5b&is=671ce9db&hm=76962ee2949d8324c2f7022ef63f8b7d3c6fe3aabf4ecf526f44249439f204ac&[/img]

Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the minimum and maximum values of $f(n)$. Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$.

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$

In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1, a_2, a_3, a_4)$ to some position $(x_1, x_2, x_3, x_4)$ with integer coordinates satisfying \[(x_1-a_1)^2 + (x_2-a_2)^2 + (x_3-a_3)^2 + (x_4-a_4)^2 = 4 \quad \text{and} \quad |(x_1 + x_2 + x_3 + x_4) - (a_1 + a_2 + a_3 + a_4)| = 2.\] In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

Call a date mm/dd/yy $\textit{multiplicative}$ if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, $01/21/21$, $03/07/21$, and $07/03/21$ are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum? $\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$