Found problems: 85335
Four numbers are written on the board: $1, 3, 6, 10.$ Each time two arbitrary numbers, $a$ and $b$ are deleted, and numbers $a + b$ and $ab$ are written in their place. Is it possible to get numbers $2015, 2016, 2017, 2018$ on the board after several such operations?
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of triangle $ACD$ intersect $AB$ again at $E$; and let the circumcircle of triangle $ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$.
Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than $\sqrt2$?
A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary.
[i]Author: Kei Irie, Japan[/i]
In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$.
[i]Proposed by Jacob Xu[/i]
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$,
(iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.
Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written?
[i]Proposed by G. Halperin[/i]
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$.
[i] Proposed by Michael Ren [/i]
$x$ is a base-$10$ number such that when the digits of $x$ are interpreted as a base-$20$ number, the resulting number is twice the value as when they are interpreted as a base-$13$ number. Find the sum of all possible values of $x$.
We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$
Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.
Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$.
[list=a]
[*]Prove that $\triangle ALE\sim\triangle AEB$.
[*]Prove that $FK=FL$.
[/list]
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
A circle $\omega$ and two points $A, B$ of this circle are given. Let $C$ be an arbitrary point on one of arcs $AB$ of $\omega$; $CL$ be the bisector of triangle $ABC$; the circle $BCL$ meet $AC$ at point $E$; and $CL$ meet $BE$ at point $F$. Find the locus of circumcenters of triangles $AFC$.
What is the value of
$$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
$\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$
A [i]mixing[/i] of the sequence $a_1,a_2,\dots ,a_{3n}$ is called the following sequence:
$a_3,a_6,\dots ,a_{3n},a_2,a_5,\dots ,a_{3n-1},a_1,a_4,\dots ,a_{3n-2}$.
Is it possible after finite amount of [i]mixings[/i] to reach the sequence $192,191,\dots ,1$ from $1,2,\dots ,192$?
Five points are on the surface of of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points?
Given a regular heptagon $A_1...A_7$. Prove that $$\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$$.
Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, prove that the area of the region enclosed by the tangents is equal to
\[\frac{AB \cdot BC \cdot CA}{2p}\]
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top?
[asy]
add(grid(7,7));
label("$\dots$", (0.5,0.5));
label("$\dots$", (1.5,0.5));
label("$\dots$", (2.5,0.5));
label("$\dots$", (3.5,0.5));
label("$\dots$", (4.5,0.5));
label("$\dots$", (5.5,0.5));
label("$\dots$", (6.5,0.5));
label("$\dots$", (1.5,0.5));
label("$\dots$", (0.5,1.5));
label("$\dots$", (0.5,2.5));
label("$\dots$", (0.5,3.5));
label("$\dots$", (0.5,4.5));
label("$\dots$", (0.5,5.5));
label("$\dots$", (0.5,6.5));
label("$\dots$", (6.5,0.5));
label("$\dots$", (6.5,1.5));
label("$\dots$", (6.5,2.5));
label("$\dots$", (6.5,3.5));
label("$\dots$", (6.5,4.5));
label("$\dots$", (6.5,5.5));
label("$\dots$", (0.5,6.5));
label("$\dots$", (1.5,6.5));
label("$\dots$", (2.5,6.5));
label("$\dots$", (3.5,6.5));
label("$\dots$", (4.5,6.5));
label("$\dots$", (5.5,6.5));
label("$\dots$", (6.5,6.5));
label("$17$", (1.5,1.5));
label("$18$", (1.5,2.5));
label("$19$", (1.5,3.5));
label("$20$", (1.5,4.5));
label("$21$", (1.5,5.5));
label("$16$", (2.5,1.5));
label("$5$", (2.5,2.5));
label("$6$", (2.5,3.5));
label("$7$", (2.5,4.5));
label("$22$", (2.5,5.5));
label("$15$", (3.5,1.5));
label("$4$", (3.5,2.5));
label("$1$", (3.5,3.5));
label("$8$", (3.5,4.5));
label("$23$", (3.5,5.5));
label("$14$", (4.5,1.5));
label("$3$", (4.5,2.5));
label("$2$", (4.5,3.5));
label("$9$", (4.5,4.5));
label("$24$", (4.5,5.5));
label("$13$", (5.5,1.5));
label("$12$", (5.5,2.5));
label("$11$", (5.5,3.5));
label("$10$", (5.5,4.5));
label("$25$", (5.5,5.5));
[/asy]
$\textbf{(A) }367 \qquad \textbf{(B) }368 \qquad \textbf{(C) }369 \qquad \textbf{(D) }379 \qquad \textbf{(E) }380$
Let $ABC$ be a triangle with $AB=5$, $BC=4$, and $CA=3$. On each side of $ABC$, externally erect a semicircle whose diameter is the corresponding side. Let $X$ be on the semicircular arc erected on side $\overline{BC}$ such that $\angle CBX$ has measure $15^\circ$. Let $Y$ be on the semicircular arc erected on side $\overline{CA}$ such that $\angle ACY$ has measure $15^\circ$. Similarly, let $Z$ be on the semicircular arc erected on side $\overline{AB}$ such that $\angle BAZ$ has measure $15^\circ$. What is the area of triangle $XYZ$?
What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\]
$\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$
Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that
[list]
[*] $a_1+a_2=b_1+b_2,$
[*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and
[*] $a_{n-1}+a_n=b_{n-1}+b_n$.
[/list]
Find the number of junk positive integers less than or equal to $2016$.
[i]Proposed by Nathan Ramesh