Found problems: 85335
For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$
If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?
An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?
For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum^{2^{12}-1}_{n=0}f(n)$$ is divided by $2^{12}$.
Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.
Let $A$, $B$, $C$, $D$ be four points on a circle in that order. Also, $AB=3$, $BC=5$, $CD=6$, and $DA=4$. Let diagonals $AC$ and $BD$ intersect at $P$. Compute $\frac{AP}{CP}$.
Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.
[i]Huang Yumin[/i]
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win?
[i]Proposed by Mykhailo Shtandenko[/i]
A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length $ L$ and width $ W$ to enclose the greatest area. Find $ L\plus{}W$.
A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)
Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]
In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).
A man walked a certain distance at a constant rate. If he had gone $\textstyle\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\textstyle\frac{1}{2}$ mile per hour slower, he would have been $2\textstyle\frac{1}{2}$ hours longer on the road. The distance in miles he walked was
$\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }15\qquad\textbf{(C) }17\frac{1}{2}\qquad\textbf{(D) }20\qquad \textbf{(E) }25$
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
Triangle $ABC$ satisfies $\angle ABC=\angle ACB=78^\circ$. Points $D$ and $E$ lie on $AB,AC$ and satisfy $\angle BCD=24^\circ$ and $\angle CBE=51^\circ$. If $\angle BED=x^\circ$, find $x$.
$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.
[u]Part 6 [/u]
[b]p16.[/b] Le[b][/b]t $p(x)$ and $q(x)$ be polynomials with integer coefficients satisfying $p(1) = q(1)$. Find the greatest integer $n$ such that $\frac{p(2023)-q(2023)}{n}$ is an integer no matter what $p(x)$ and $q(x)$ are.
[b]p17.[/b] Find all ordered pairs of integers $(m,n)$ that satisfy $n^3 +m^3 +231 = n^2m^2 +nm.$
[b]p18.[/b] Ben rolls the frustum-shaped piece of candy (shown below) in such a way that the lateral area is always in contact with the table. He rolls the candy until it returns to its original position and orientation. Given that $AB = 4$ and $BD =CD = 3$, find the length of the path traced by $A$.
[u]Part 7 [/u]
[b]p19.[/b] In their science class, Adam, Chris, Eddie and Sam are independently and randomly assigned an integer grade between $70$ and $79$ inclusive. Given that they each have a distinct grade, what is the expected value of the maximum grade among their four grades?
[b]p20.[/b] Let $ABCD$ be a regular tetrahedron with side length $2$. Let point $E$ be the foot of the perpendicular
from $D$ to the plane containing $\vartriangle ABC$. There exist two distinct spheres $\omega_1$ and $\omega_2$, centered at points $O_1$ and $O_2$ respectively, such that both $O_1$ and $O_2$ lie on $\overrightarrow{DE}$ and both spheres are tangent to all four of the planes $ABC$, $BCD$, $CDA$, and $DAB$. Find the sum of the volumes of $\omega_1$ and $\omega_2$.
[b]p21.[/b] Evaluate
$$\sum^{\infty}_{i=0}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0} \frac{1}{(i + j +k +1)2^{i+j+k+1}}.$$
[u]Part 8 [/u]
[b]p22.[/b] In $\vartriangle ABC$, let $I_A$, $I_B$ , and $I_C$ denote the $A$, $B$, and $C$-excenters, respectively. Given that $AB = 15$, $BC = 14$ and $C A = 13$, find $\frac{[I_A I_B I_C ]}{[ABC]}$ .
[b]p23.[/b] The polynomial $x +2x^2 +3x^3 +4x^4 +5x^5 +6x^6 +5x^7 +4x^8 +3x^9 +2x^{10} +x^{11}$ has distinct complex roots $z_1, z_2, ..., z_n$. Find $$\sum^n_{k=1} |R(z^2n))|+|I(z^2n)|,$$ where $R(z)$ and $I(z)$ indicate the real and imaginary parts of $z$, respectively. Express your answer in simplest radical form.
[b]p24.[/b] Given that $\sin 33^o +2\sin 161^o \cdot \sin 38^o = \sin n^o$ , compute the least positive integer value of $n$.
[u]Part 9[/u]
[b]p25.[/b] Submit a prime between $2$ and $2023$, inclusive. If you don’t, or if you submit the same number as another team’s submission, you will receive $0$ points. Otherwise, your score will be $\min \left(30, \lfloor 4 \cdot ln(x) \rfloor \right)$, where $x$ is the positive difference between your submission and the closest valid submission made by another team.
[b]p26.[/b] Sam, Derek, Jacob, andMuztaba are eating a very large pizza with $2023$ slices. Due to dietary preferences, Sam will only eat an even number of slices, Derek will only eat a multiple of $3$ slices, Jacob will only eat a multiple of $5$ slices, andMuztaba will only eat a multiple of $7$ slices. How many ways are there for Sam, Derek, Jacob, andMuztaba to eat the pizza, given that all slices are identical and order of slices eaten is irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be:
$$\max \left( 0, \left\lfloor 30 \left( 1-2\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$
[b]p27.[/b] Let $ \Omega_(k)$ denote the number of perfect square divisors of $k$. Compute $$\sum^{10000}_{k=1} \Omega_(k).$$
If your answer is $A$ and the correct answer is $C$, the number of points you recieve will be
$$\max \left( 0, \left\lfloor 30 \left( 1-4\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$
PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3267911p30056982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that
\[
xf(x+y)=(x+\alpha y)f(x)+xf(y)
\]
for all $x,y\in\mathbb{R}$.
Triangle $ABC$ has circumcenter $O$ and circumcircle $\omega$. Let $A_{\omega}$ be the point diametrically opposite $A$ on $\omega$, and let $H$ be the foot of the altitude from $A$ onto $BC$. Let $H_B$ and $H_C$ be the reflections of $H$ over $B$ and $C$, respectively. Point $P$ is the intersection of line $A_{\omega}B$ and the perpendicular of $BC$ at point $H_B$, and point $Q$ is the intersection of line $A_{\omega}C$ and the perpendicular of $CB$ at point $H_C$. The circles $\omega_1$ and $\omega_2$ have the respective centers $P$ and $Q$ and respective radii $PA$ and $QA$. Suppose that $\omega$, $\omega_1$, and $\omega_2$ intersect at another common point $X$. If $AO = \frac{\sqrt{105}}{5}$ and $AX = 4$, then $|AB - CA|^2$ can be written as $m - n\sqrt{p}$ for positive integers $m$ and $n$ and squarefree positive integer $p$. Find $m + n + p$.
[i]Note: the reflection of a point $P$ over another point $Q \neq P$ is the point $P'$ such that $Q$ is the midpoint of $P$ and $P'$.[/i]
Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[color=darkred]Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ .
[b]a)[/b] Prove that : $AB=BA$ .
[b]b)[/b] Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .[/color]
Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.
$ABCD$ is a cyclic quadrilateral with $\angle BAD=90^\circ$ and $\angle ABC>90^\circ$. $AB$ is extended to a point $E$ such that $\angle AEC=90^\circ$.If $AB=7,BE=9,$ and $EC=12$,calculate $AD$.
Let $\omega \ne 1$ be a $13$th root of unity. Find the remainder when \[ \prod_{k=0}^{12} \left(2 - 2\omega^k + \omega^{2k} \right) \] is divided by $1000$.
a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` Now I can tell what they are.'
What are the two numbers?
b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?