If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$. [i]Serbia[/i]

Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let\[ S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)} \]Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: \[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]

Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.

Points on complex plane that complex numbers $z_1,z_2$ corresponding to are $A,B$, and $|z_1|=4,4z_1^2-2z_1z_2+z_2^2=0$. $O$ is original point, then the area of $\triangle OAB$ is $\text{(A)}8\sqrt3\qquad\text{(B)}4\sqrt3\qquad\text{(C)}6\sqrt3\qquad\text{(D)}12\sqrt3$

For a sequence $u_1,u_2\dots,$ define $\Delta^1(u_n)=u_{n+1}-u_n$ and, for all integer $k>1$, $\Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n))$. If $u_n=n^3+n$, then $\Delta^k(u_n)=0$ for all $n$ $\textbf{(A) }\text{if }k=1\qquad$ $\textbf{(B) }\text{if }k=2,\text{ but not if }k=1\qquad$ $\textbf{(C) }\text{if }k=3,\text{ but not if }k=2\qquad$ $\textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad$ $\textbf{(E) }\text{for no value of }k$

Two of the Geometry box tools are placed on the table as shown. Determine the angle $\angle ABC$ [img]https://2.bp.blogspot.com/--DWVwVQJgMM/XU1OK08PSUI/AAAAAAAAKfs/dgZeYwiYOrQJE4eKQT5s13GQdBEHPqy9QCK4BGAYYCw/s1600/prmo%2B16%2BChandigarh%2Bp7.png[/img]

Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$ [i]Proposed by Mihai Baluna, Romania[/i]

Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that \[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

The figure shows a network of roads bounding $12$ blocks. Person $P$ goes from $A$ to $B,$ and person $Q$ goes from $B$ to $A,$ each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. Find the probability that the persons meet. [asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; draw((0,3)--(4,3),linewidth(1.2pt)); draw((4,3)--(4,0),linewidth(1.2pt)); draw((4,0)--(0,0),linewidth(1.2pt)); draw((0,0)--(0,3),linewidth(1.2pt)); draw((1,3)--(1,0),linewidth(1.2pt)); draw((2,3)--(2,0),linewidth(1.2pt)); draw((3,3)--(3,0),linewidth(1.2pt)); draw((0,1)--(4,1),linewidth(1.2pt)); draw((4,2)--(0,2),linewidth(1.2pt)); dot((0,0),ds); label("$A$", (-0.3,-0.36),NE*lsf); dot((4,3),ds); label("$B$", (4.16,3.1),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]

Prove, that for every natural $N$ exists $k$, such that $N=a_02^0+a_12^1+...+a_k2^k$, where $a_0,a_1,...a_k$ are $1$ or $2$

Let $a_1, a_2, \ldots, a_{11}$ be integers. Prove that there exist numbers $b_1, b_2, \ldots, b_{11}$ such that [list] [*] $b_i$ is equal to $-1,0$ or $1$ for all $i \in \{1, 2,\dots, 11\}$. [*] all numbers can't be zero at a time. [*] the number $N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}$ is divisible by $2024$. [/list]

A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows: i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread; ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors. Show that we can assign to each node on the ball a complex number (not equal to $1$), such that the product of the numbers assigned to the vertices of any polygonal face is equal to $1$.

Suppose $a$ and $b$ are positive integers such that $a^b=2^{2023}.$ Compute the smallest possible value of $b^a.$

The radius of the circumscribed circle of an acute-angled triangle is $23$ and the radius of its Inscribed circle is $9$. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.

let n,m be positive integers st $m>n\geq 5$ with m depending on n. consider the sequence $a_1,a_2,...a_m$ where $a_i=i$ for $i=1,...,n$ $a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}$ for $j=1,..,m-n$ with $m-3(m-n)=$1 or 2, ie $a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2}$ where k=1 or 2 (Thus if $n=5$, the sequence is 1,2,3,4,5,6,15 and if $n=8$, the sequence is 1,2,3,4,5,6,7,8,6,15,21) Find $S=a_1+...+a_m$ if (i) $n=2007$ (ii) $n=2008$

If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula \[ \binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, . \] Compute the value of \[ \frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, . \]

Which of the following is equal to $ \dfrac{\frac{1}{3}\minus{}\frac{1}{4}}{\frac{1}{2}\minus{}\frac{1}{3}}$? $ \textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{3}{4}$

Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential. [asy] import roundedpath; size(300); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0.102, 6.163)-- (0.192, 6.358)-- (0.369, 6.500)-- (0.526, 6.642)-- (0.643, 6.712)-- (0.820, 6.830)-- (0.938, 6.901)-- (1.075, 7.043)-- (1.193, 7.185)-- (1.369, 7.256)-- (1.506, 7.374)-- (1.644, 7.445)-- (1.840, 7.515)-- (1.958, 7.586)-- (2.134, 7.657)-- (2.291, 7.752)-- (2.468, 7.846)-- (2.625, 7.846)-- (2.899, 7.893)-- (3.095, 8.035)-- (3.350, 8.035)-- (3.586, 8.106)-- (3.860, 8.106)-- (4.135, 8.106)-- (4.371, 8.035)-- (4.606, 8.035)-- (4.881, 8.012)-- (5.155, 7.917)-- (5.391, 7.823)-- (5.665, 7.728)-- (5.960, 7.563)-- (6.175, 7.468)-- (6.332, 7.374)-- (6.528, 7.232)-- (6.725, 7.161)-- (6.882, 6.996)-- (7.117, 6.854)-- (7.333, 6.712)-- (7.509, 6.523)-- (7.666, 6.358)-- (7.902, 6.146)-- (8.098, 5.980)-- (8.274, 5.791)-- (8.451, 5.649)-- (8.647, 5.484)-- (8.882, 5.248)-- (9.196, 5.059)-- (9.392, 4.894)-- (9.628, 4.752)-- (9.824, 4.634)-- (10.118, 4.516)-- (10.452, 4.350)-- (10.785, 4.232)-- (11.001, 4.185)-- (11.315, 4.138)-- (11.648, 4.114)-- (12.002, 4.114)-- (12.257, 4.091)-- (12.610, 4.067)-- (12.825, 4.161)-- (13.081, 4.185)-- (13.316, 4.279)-- (13.492, 4.327)-- (13.689, 4.445)-- (13.826, 4.516)-- (14.022, 4.587)-- (14.159, 4.705)-- (14.316, 4.823)-- (14.532, 4.964)-- (14.669, 5.059)-- (14.866, 5.177)-- (15.062, 5.248)-- (15.278, 5.461)-- (15.474, 5.697)-- (15.650, 5.838)-- (15.847, 6.004)-- (16.043, 6.169)-- (16.258, 6.334)-- (16.415, 6.523)-- (16.592, 6.736)-- (16.788, 6.830)-- (17.063, 7.067)-- (17.357, 7.232)-- (17.573, 7.397)-- (17.808, 7.515)-- (18.063, 7.634)-- (18.358, 7.704)-- (18.573, 7.870)-- (18.887, 7.941)-- (19.142, 8.012)-- (19.358, 8.035)-- (19.574, 8.082)-- (19.770, 8.130); draw(shift(1.8*up)*roundedpath(A,0.09),linewidth(1.5)); [/asy] What is the total energy of the particle? (A) $\text{-5 J}$ (B) $\text{0 J}$ (C) $\text{5 J}$ (D) $\text{10 J}$ (E) $\text{15 J}$

Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.