Found problems: 85335
A finite set $M$ of unit squares on the plane is considered. The sides of the squares are parallel to the coordinate axes and the squares are allowed to intersect. It is known that the distance between the centres of any pair of squares is no greater than $2$. Prove that there exists a unit square (not necessarily belonging to $M$) with sides parallel to the coordinate axes and which has at least one common point with each of the squares in $M$.
(A Andjans, Riga)
Prove that
$$\int^{\frac\pi3}_{\frac\pi6}\frac u{\sin u}du=\frac83\sum_{k=0}^\infty\frac{(-1)^k}{3^k(2k+1)^2}+\frac{\pi\ln3}{3\sqrt3}-\frac{4C}3+\frac\pi6\ln\left(2+\sqrt3\right)-\operatorname{Im}\left(\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)-\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)$$
where as usual
$$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$
and $C=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}$ is the Catalan constant.
[i]Proposed by Paolo Perfetti[/i]
Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $a_1\leq a_2 \leq \cdots \leq a_n .$ Prove that
$$\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.$$
[b]p1.[/b] If $n$ is a positive integer such that $2n+1 = 144169^2$, find two consecutive numbers whose squares add up to $n + 1$.
[b]p2.[/b] Katniss has an $n$-sided fair die which she rolls. If $n > 2$, she can either choose to let the value rolled be her score, or she can choose to roll a $n - 1$ sided fair die, continuing the process. What is the expected value of her score assuming Katniss starts with a $6$ sided die and plays to maximize this expected value?
[b]p3.[/b] Suppose that $f(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, and that $f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 7$. What is $a$?
[b]p4.[/b] $a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$.
[b]p5.[/b] Square $ABCD$ and rhombus $CDEF$ share a side. If $m\angle DCF = 36^o$, find the measure of $\angle AEC$.
[b]p6.[/b] Tom challenges Harry to a game. Tom first blindfolds Harry and begins to set up the game. Tom places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules:
(a) Harry is allowed to flip as many coins as he wants during his turn.
(b) A turn consists of Harry flipping as many coins as he wants (while blindfolded). When he is happy with what he has flipped, Harry will ask Tom whether or not he was successful in flipping all the coins face-up or face-down. If yes, then Harry wins. If no, then Tom will take the index card back, rotate the card however he wants, and return it back to Harry, thus starting Harry’s next turn. Note that Tom cannot touch the coins after he initially places them before starting the game.
Assuming that Tom’s initial configuration of the coins weren’t all face-up or face-down, and assuming that Harry uses the most efficient algorithm, how many moves maximum will Harry need in order to win? Or will he never win?
PS. You had better use hide for answers.
Given a list of integers $2^1+1, 2^2+1, \ldots, 2^{2019}+1$, Adam chooses two different integers from the list and computes their greatest common divisor. Find the sum of all possible values of this greatest common divisor.
We have an integer $A$ such that $A^2$ is a four digit number, with $5$ in the ten's place . Find all possible values of $A$.
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins.
Find if one of the players has a winning strategy, and if so which player has a winning strategy.
[i]Théo Lenoir, France[/i]
Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if
\[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]
Directed lines are drawn from the center of a circle, making angles of $0, \pm 1, \pm 2, \pm 3, \ldots$ (measured in radians from a prime direction). If these lines meet the circle in points $P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots,$ show that there is no interval on the circumference of the circle which does not contain some $P_{\pm i}.$ (You may assume that $\pi$ is irrational.)
There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.
Find all non-negative integers $n$ for which the equation
\[
{\left( x^2 + y^2 \right)}^n = {(xy)}^{2018}
\]
admits positive integral solutions.
Evaluate the expression
\[
\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
11 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, .
\]
Let $P$ be a point on the circumcircle of triangle $ABC$. Construct an arbitrary triangle $A_1B_1C_1$ whose sides $A_1B_1$, $B_1C_1$ and $C_1A_1$ are parallel to the segments $PC$, $PA$ and $PB$ respectively and draw lines through the vertices $A_1$, $B_1$ and $C_1$ and parallel to the sides $BC$, $CA$ and $AB$ respectively. Prove that these three lines have a common point lying on the circumcircle of triangle $A_1B_1C_1$.
(V. Prasolov)
If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation
$\textbf{(A) }r^2+r+1=0\qquad\textbf{(B) }r^2-r+1=0\qquad\textbf{(C) }r^4+r^2-1=0$
$\qquad\textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$
In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD.$ The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ,$ for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$
[asy]
import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen fftttt = rgb(1,0.2,0.2);
draw(circle((6.04,2.8),1.78),qqttff); draw((6.02,4.58)--(6.04,2.8),fftttt); draw((6.02,4.58)--(6.98,4.56),fftttt); draw((6.04,2.8)--(8.13,2.88),fftttt); draw((6.98,4.56)--(8.13,2.88),fftttt);
dot((6.04,2.8),ds); label("$B$", (5.74,2.46), NE*lsf); dot((6.02,4.58),ds); label("$A$", (5.88,4.7), NE*lsf); dot((6.98,4.56),ds); label("$F$", (7.06,4.6), NE*lsf); dot((7.39,3.96),ds); label("$E$", (7.6,3.88), NE*lsf); dot((8.13,2.88),ds); label("$D$", (8.34,2.56), NE*lsf); dot((7.82,2.86),ds); label("$C$", (7.5,2.46), NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);
[/asy]
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$
Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.
Some number of regular polygons meet at a point on the plane such that the polygons' interiors do not overlap, but the polygons fully surround the point (i.e. a sufficiently small circle centered at the point would be contained in the union of the polygons). What is the largest possible number of sides in any of the polygons?
Given $\triangle ABC$ with $AB < AC$, let $\omega$ be the circle centered at the midpoint $M$ of $BC$ with diameter $AC - AB$. The internal bisector of $\angle BAC$ intersects $\omega$ at distinct points $X$ and $Y$. Let $T$ be the point on the plane such that $TX$ and $TY$ are tangent to $\omega$. Prove that $AT$ is perpendicular to $BC$.
Find all pairs $(a,b)$ of positive integers, such that for [b]every[/b] $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.
Using only prime numbers, a set is formed with the following conditions:
Any one-digit prime number can be in the set.
For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set.
Write, of the sets that meet these conditions, the one with the greatest number of elements.
Justify why there cannot be one with more elements.
Remember that the number $1$ is not prime.
In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} \equal{} 3: 3: 1$, and similarly for lines $ BE$ and $ CF.$ Then the area of triangle $ N_1N_2N_3$ is:
[asy]unitsize(27);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair A,B,C,D,E,F,X,Y,Z;
A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1);
draw(A--B--C--cycle);
draw(A--D); draw(B--E); draw(C--F);
X=intersectionpoint(A--D,C--F); Y=intersectionpoint(B--E,A--D); Z=intersectionpoint(B--E,C--F);
label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE);
label("$D$",D,S); label("$E$",E,NE); label("$F$",F,NW);
label("$N_1$",X,NE); label("$N_2$",Y,WNW); label("$N_3$",Z,S);[/asy]$ \textbf{(A)}\ \frac {1}{10} \triangle ABC \qquad\textbf{(B)}\ \frac {1}{9} \triangle ABC \qquad\textbf{(C)}\ \frac {1}{7} \triangle ABC \qquad\textbf{(D)}\ \frac {1}{6} \triangle ABC \qquad\textbf{(E)}\ \text{none of these}$
The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$.
Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$.
[img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]