Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality \[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]

Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$? $ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad $

$\triangle ABC$ is a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ is the bisector of $\angle ABC$, then $\angle BDC =$ [asy] size(200); defaultpen(linewidth(0.8)+fontsize(11pt)); pair A= origin, B = 3 * dir(25), C = (B.x,0); pair X = bisectorpoint(A,B,C), D = extension(B,X,A,C); draw(B--A--C--B--D^^rightanglemark(A,C,B,4)); path g = anglemark(A,B,D,14); path h = anglemark(D,B,C,14); draw(g); draw(h); add(pathticks(g,1,0.11,6,6)); add(pathticks(h,1,0.11,6,6)); label("$A$",A,W); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,S); label("$20^\circ$",A,8*dir(12.5)); [/asy] $ \textbf{(A)}\ 40^\circ \qquad \textbf{(B)}\ 45^\circ \qquad \textbf{(C)}\ 50^\circ \qquad \textbf{(D)}\ 55^\circ \qquad \textbf{(E)}\ 60^\circ $

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

There are given 11 distinct points inside a ball with volume $V.$ Show that there are two planes $\varrho,\sigma,$ both containing the center of the ball, such that the resulting spherical wedge has volume $V/8$ and its interior contains none of the given points.

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $

Let $(t_n)_{n\geqslant 1}$ be the sequence defined by $t_1=1, t_{2k}=-t_k$ and $t_{2k+1}=t_{k+1}$ for all $k\geqslant 1.$ Consider the series \[\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.\]Prove that this series converges to a positive real number. [i]Proposed by Dylan Toh[/i]

If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$ [u]Babis[/u] [b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]

Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it? $ \text{(A)}\ 5724\qquad\text{(B)}\ 7245\qquad\text{(C)}\ 7254\qquad\text{(D)}\ 7425\qquad\text{(E)}\ 7542 $

We are given a graph $G$ with $n$ edges. For each edge, we write down the lesser degree of two vertices at the end of that edge. Prove that the sum of the resulting $n$ numbers is at most $100n\sqrt{n}$.

What is the value of \[\left(9+\dfrac{9}{9}\right)^{9-9/9} - \dfrac{9}{9}?\] [i]Proposed by David Altizio[/i]

A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.

If the sequence $\{a_n\}$ is defined by \begin{align*}a_1 &= 2,\\ a_{n+1} &= a_n + 2n\qquad (n\geq 1),\end{align*} then $a_{100}$ equals $\textbf{(A) }9900\qquad \textbf{(B) }9902\qquad \textbf{(C) }9904\qquad \textbf{(D) }10100\qquad \textbf{(E) }10102$

Define the function $m$ of the three real variables $x$, $y$, $z$ by $m$($x$,$y$,$z$) = max($x^2$,$y^2$,$z^2$), $x$, $y$, $z$ ∈ $R$. Determine, with proof, the minimum value of $m$ if $x$,$y$,$z$ vary in $R$ subject to the following restrictions: $x$ + $y$ + $z$ = 0, $x^2$ + $y^2$ + $z^2$ = 1.

A cube is divided into $27$ unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these $8$ spheres. What is the smallest possible value for the radius of the last sphere?

It is possible to express the sum $$\sum_{n=1}^{24}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$ as $a\sqrt{2}+b\sqrt{3}$, for some integers $a$ and $b$. Compute the ordered pair $(a,b)$.

[b]6.[/b] For which Lebesgue-measurable subsets $E$ of the real line does a positive constant $c$ exist for which $\sup_{-\infty < t<\infty} \left | \int_{E} e^{itx} f(x) dx\right | \leq c \sup_{n=0,\pm 1,\dots } \left | \int_{E} e^{inx} f(x) dx\right |$ for all integrable functions $f$ on $E$? ([b]M.17[/b]) [G. Halász]

$S$ is a subset of Natural numbers which has infinite members. $$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$ Prove the set of prime divisors of $S’$ has also infinite members [i]Proposed by Yahya Motevassel[/i]

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$. [asy] import three; size(250); defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2"); currentprojection = orthographic(0,-15,5); draw(circle((0,0,0), 15),dashes); draw(circle((0,0,80), 15)); draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80)); draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes); draw("48", (-24,0,-20)--(24,0,-20)); draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17)); label("30", (0,0,-15)); draw("60", (50,0,0)--(50,0,60)); draw("20", (50,0,60)--(50,0,80)); draw((50,0,60)--(47,0,60));[/asy]

Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that $\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$. Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$. 

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10\%$. She leaves a $15\%$ tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ for dinner. What is the cost of here dinner without tax or tip? $ \textbf{(A)}\ \$18\qquad\textbf{(B)}\ \$20\qquad\textbf{(C)}\ \$21\qquad\textbf{(D)}\ \$22\qquad\textbf{(E)}\ \$24$