Found problems: 85335
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$
where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.
[i]V. Petkov[/i]
[u]Round 1[/u]
[b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$?
[b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img]
[b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img]
[b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken.
[img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img]
[b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$?
[u]Round 2[/u]
[b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img]
Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img]
[b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$.
Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$.
[i]1997 Hokkaido University entrance exam/Science[/i]
Each student chooses $1$ math problem and $1$ physics problem among $20$ math problems and $11$ physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?
Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$
The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.
On the grid plane all possible broken lines with the following properties are constructed:
each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$.
(Ilke Chanakchi, Ralf Schiffler)
$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that
$$ TB\cdot TC=TI_b^2.$$
For every $n \in \mathbb{N}$, there exist integers $k$ such that $n | k$ and $k$ contains only zeroes and ones in its decimal representation. Let $f(n)$ denote the least possible number of ones in any such $k$. Determine whether there exists a constant $C$ such that $f(n) < C$ for all $n \in \mathbb{N}$.
Let \( f(x) = \frac{100^x}{100^x + 10} \). Determine the value of:
\[
f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right)
\]
A box $3\times5\times7$ is divided into unit cube cells. In each of the cells, there is a c[i][/i]ockchafer. At a signal, every c[i][/i]ockchafer moves through a face of its cell to a neighboring cell.
(a) What is the minimum number of empty cells after the signal?
(b) The same question, assuming that the c[i][/i]ockchafers move to diagonally adjacent cells (sharing exactly one vertex).
You are given a word consisting of letters $a;b;c$ You can apply 3 operations on this word:
[b]1)[/b] You can write any $3$ letter long $\text{subword}$ in reverse. ($\text{xyz}\rightarrow \text{zyx}$)
[b]2)[/b] You can add same $2$ letters between any $2$ consecutive letters. ($\text{xyxy}\rightarrow \text{xy}$[b]zz[/b]$\text{xy}$)
[b]3)[/b] You can remove any $\text{subword}$ in the given form $\text{xyyx}$ ($\text{x}$[b]yzzy[/b]$\text{xy}\rightarrow\text{xxy}$)
Given these $3$ operations, can you go from $\text{abccab}$ to $\text{baccba}$?
(Note: A $\text{subword}$ is a string of consecutive letters from the given word)
Solve the system of equations $\begin{cases} x^3+y = x^2+1\\
2y^3+z=2y^2+1 \\
3z^3+x=3z^2+1 \end{cases}$
[center][u]Guts Round[/u] / [u]Set 7[/u][/center]
[b]p19.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21.
Compute the real value of $a$ such that $$\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.$$
[b]p20.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21.
For some triangle $\vartriangle ABC$, let $\omega$ and $\omega_A$ be the incircle and $A$-excircle with centers $I$ and $I_A$, respectively. Suppose $AC$ is tangent to $\omega$ and $\omega_A$ at $E$ and $E'$, respectively, and $AB$ is tangent to $\omega$ and $\omega_A$ at $F$ and $F'$ respectively. Furthermore, let $P$ and $Q$ be the intersections of $BI$ with $EF$ and $CI$ with $EF$, respectively, and let $P'$ and $Q'$ be the intersections of $BI_A$ with $E'F'$ and $CI_A$ with $E'F'$, respectively. Given that the circumradius of $\vartriangle ABC$ is a, compute the maximum integer value of $BC$ such that the area $[P QP'Q']$ is less than or equal to $1$.
[b]p21.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21.
Let $c$ be a positive integer such that $gcd(b, c) = 1$. From each ordered pair $(x, y)$ such that $x$ and $y$ are both integers, we draw two lines through that point in the $x-y$ plane, one with slope $\frac{b}{c}$ and one with slope $-\frac{c}{b}$ . Given that the number of intersections of these lines in $[0, 1)^2$ is a square number, what is the smallest possible value of $ c$?
Note that $[0, 1)^2$ refers to all points $(x, y)$ such that $0 \le x < 1$ and $ 0 \le y < 1$.
There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.
The annual incomes of $1000$ families range from $8200$ dollars to $98000$ dollars. In error, the largest income was entered on the computer as $980000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is
$ \text{(A)}\ \text{882 dollars}\qquad\text{(B)}\ \text{980 dollars}\qquad\text{(C)}\ \text{1078 dollars}\qquad\text{(D)}\ \text{482,000 dollars}\qquad\text{(E)}\ \text{882,000 dollars} $
Let
$$N = {2^{2012} \choose 0} {2^{2012} \choose 1} {2^{2012} \choose 2} {2^{2012} \choose 3}... {2^{2012} \choose 2^{2012}}.$$
Let M be the number of $0$’s when $N$ is written in binary. How many $0$’s does $M$ have when written in binary?
(Warning: this question is very hard.)
In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$
\[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \]
where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Show that $L_{2n+1}+(-1)^{n+1}(n \geq 1)$ can be written as a product of three (not necessarily distinct) Fibonacci numbers.
Let $a$ be a positive number, and we show decimal part of the $a$ with $\left\{a\right\}$.For a positive number $x$ with $\sqrt 2< x <\sqrt 3$ such that, $\left\{\frac{1}{x}\right\}$=$\left\{x^2\right\}$.Find value of the
$$x(x^7-21)$$
Let $I$ be a point on the bisector of angle $BAC$ of a triangle $ABC$. Points $M,N$ are taken on the respective sides $AB$ and $AC$ so that $\angle ABI=\angle NIC$ and $\angle ACI=\angle MIB$. Show that $I$ is the incenter of triangle $ABC$ if and only if points $M,N$ and $I$ are collinear.
Let $A_1A_2A_3A_4A_5A_6$ be a convex hexagon. Suppose that there exists 2 points $P,Q$ in its interior such that $\angle A_{i-1}A_iP=\angle QA_iA_{i+1}$ for $i=1,2,\ldots,6$ where $A_0\equiv A_6,A_1\equiv A_7$. Prove that
\[\angle A_1PA_2+\angle A_3PA_4+\angle A_5PA_6=180^\circ.\]
A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$, $0 <d <1$. A line through $A$ and $P$ intersects the semicircle at point $C$. Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$. Through point $C$ we draw a line $l_1$, parallel to $PD$ and then a line $l_2$, through $D$ parallel to $PC$. The lines $l_1$ and $l_2$ intersect at point $E$. Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$