SEEMOUS 2016 COMPETITION PROBLEMS

We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.

In a convex quadrilateral $ABCD$, the lines $AB$ and $DC$ intersect at point $P{}$ and the lines $AD$ and $BC$ intersect at point $Q{}$. The points $E{}$ and $F{}$ are inside the quadrilateral $ABCD$ such that the circles $(ABE), (CDE), (BCF),(ADF)$ intersect at one point $K{}$. Prove that the circles $(PKF)$ and $(QKE)$ intersect a second time on the line $PQ$.

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)

Compute $99(99^2+3) + 3\cdot99^2$. [i]Proposed by Evan Chen[/i]

In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D); label("$A$", A, dir(0)); label("$B$", B, dir(180)); label("$C$", C, dir(110)); label("$D$", D, dir(130)); [/asy] $ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$

$m$ and $n$ are positive integers. If the number \[ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}\] is an integer, prove that $k$ is a perfect square.

There are $n$ cells with indices from $1$ to $n$. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the $i$-th cell is now a card with the number $a_i$. Petya can swap any two cards with the numbers $x$ and $y$, but he must pay $2|x-y|$ coins. Show that Petya can return all the cards to their original position, not paying more than $|a_1-1|+|a_2-2|+\ldots +|a_n-n|$ coins.

Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.

Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

Three children wanted to make a table-game. For that purpose they wished to enumerate the $mn$ squares of an $m \times n$ game-board by the numbers $1, ... ,mn$ in such way that the numbers $1$ and $mn$ lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number $1$) in one of the corners but each child wanted to have the final square (with number $mn$ ) in different corner. For which numbers $m$ and $n$ is it possible to satisfy the wish of any of the children?

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..

Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer. [i]Proposed by Prajit Adhikari, Nepal[/i]

Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.

Let $n$ be a positive integer. If the number $1998$ is written in base $n$, a three-digit number with the sum of digits equal to $24$ is obtained. Find all possible values of $n$.

Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties: $\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order. $\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)

Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made? [asy] unitsize(1 cm); real unitwidth, dayheight, barheight; int i; unitwidth = 0.5; dayheight = 1; barheight = 0.3; draw((unitwidth,0)--(unitwidth,5*dayheight),gray(0.7)); draw((2*unitwidth,0)--(2*unitwidth,5*dayheight),gray(0.7)); draw((3*unitwidth,0)--(3*unitwidth,5*dayheight),gray(0.7)); draw((4*unitwidth,0)--(4*unitwidth,5*dayheight),gray(0.7)); draw((5*unitwidth,0)--(5*unitwidth,5*dayheight),gray(0.7)); draw((6*unitwidth,0)--(6*unitwidth,5*dayheight),gray(0.7)); draw((7*unitwidth,0)--(7*unitwidth,5*dayheight),gray(0.7)); fill((0,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight + 1/2*barheight)--(0,1/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); fill((0,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight + 1/2*barheight)--(0,5/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((8*unitwidth,0)--(8*unitwidth,5*dayheight),gray(0.7)); draw((9*unitwidth,0)--(9*unitwidth,5*dayheight),gray(0.7)); fill((0,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight + 1/2*barheight)--(0,9/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((10*unitwidth,0)--(10*unitwidth,5*dayheight),gray(0.7)); fill((0,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight + 1/2*barheight)--(0,3/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((11*unitwidth,0)--(11*unitwidth,5*dayheight),gray(0.7)); draw((12*unitwidth,0)--(12*unitwidth,5*dayheight),gray(0.7)); fill((0,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight + 1/2*barheight)--(0,7/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((0*unitwidth,0)--(0*unitwidth,5*dayheight),gray(0.7)); draw((13*unitwidth,0)--(13*unitwidth,5*dayheight),gray(0.7)); draw((14*unitwidth,0)--(14*unitwidth,5*dayheight),gray(0.7)); label("$0$", (0,5*dayheight), N); label("$4$", (2*unitwidth,5*dayheight), N); label("$8$", (4*unitwidth,5*dayheight), N); label("$12$", (6*unitwidth,5*dayheight), N); label("$16$", (8*unitwidth,5*dayheight), N); label("$20$", (10*unitwidth,5*dayheight), N); label("$24$", (12*unitwidth,5*dayheight), N); label("$28$", (14*unitwidth,5*dayheight), N); label("Number of students at soccer practice", (7*unitwidth,6*dayheight)); label("Monday", (-0.5*unitwidth,9/2*dayheight), W); label("Tuesday", (-0.5*unitwidth,7/2*dayheight), W); label("Wednesday", (-0.5*unitwidth,5/2*dayheight), W); label("Thursday", (-0.5*unitwidth,3/2*dayheight), W); label("Friday", (-0.5*unitwidth,1/2*dayheight), W); [/asy] $\textbf{(A) } \text{The mean increases by 1 and the median does not change.}$ $\textbf{(B) } \text{The mean increases by 1 and the median increases by 1.}$ $\textbf{(C) } \text{The mean increases by 1 and the median increases by 5.}$ $\textbf{(D) } \text{The mean increases by 5 and the median increases by 1.}$ $\textbf{(E) } \text{The mean increases by 5 and the median increases by 5.}$

Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$

We call a natural number venerable if the sum of all its divisors, including $1$, but not including the number itself, is $1$ less than this number. Find all the venerable numbers, some exact degree of which is also venerable.

Three identical circles are packed into a unit square. Each of the three circles are tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\tfrac{1}{r})^2$?