Found problems: 85335
Novosibirsk Oral Geo Oly VIII, 2021.4
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.
1998 Irish Math Olympiad, 2
Prove that if $ a,b,c$ are positive real numbers, then:
$ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$
1998 All-Russian Olympiad, 2
A convex polygon is partitioned into parallelograms. A vertex of the polygon is called [i]good[/i] if it belongs to exactly one parallelogram. Prove that there are more than two good vertices.
2021 Iran MO (3rd Round), 1
For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.
2006 Tournament of Towns, 2
Prove that one can find 100 distinct pairs of integers such that every digit of each number is no less than 6 and the product of the numbers in each pair is also a number with all its digits being no less than 6.
[i](4 points)[/i]
2014 NIMO Problems, 9
This is an ARML Super Relay! I'm sure you know how this works! You start from #1 and #15 and meet in the middle.
We are going to require you to solve all $15$ problems, though -- so for the entire task, submit the sum of all the answers, rather than just the answer to #8.
Also, uhh, we can't actually find the slip for #1. Sorry about that. Have fun anyways!
Problem 2.
Let $T = TNYWR$. Find the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy.
Problem 3.
Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$.
Problem 4.
Let $T = TNYWR$ and flip $4$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Problem 5.
Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$.
Problem 6.
Let $T = TNYWR$ and flip $6$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Problem 7.
Let $T = TNYWR$. Compute the smallest prime $p$ for which $n^T \not\equiv n \pmod{p}$ for some integer $n$.
Problem 8.
Let $M$ and $N$ be the two answers received, with $M \le N$. Compute the number of integer quadruples $(w,x,y,z)$ with $w+x+y+z = M \sqrt{wxyz}$ and $1 \le w,x,y,z \le N$.
Problem 9.
Let $T = TNYWR$. Compute the smallest integer $n$ with $n \ge 2$ such that $n$ is coprime to $T+1$, and there exists positive integers $a$, $b$, $c$ with $a^2+b^2+c^2 = n(ab+bc+ca)$.
Problem 10.
Let $T = TNYWR$ and flip $10$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Problem 11.
Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$.
Problem 12.
Let $T = TNYWR$ and flip $12$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Problem 13.
Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$.
Problem 14.
Let $T = TNYWR$. Compute the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy.
Also, we can't find the slip for #15, either. We think the SFBA coaches stole it to prevent us from winning the Super Relay, but that's not going to stop us, is it? We have another #15 slip that produces an equivalent answer. Here you go!
Problem 15.
Let $A$, $B$, $C$ be the answers to #8, #9, #10. Compute $\gcd(A,C) \cdot B$.
1967 Leningrad Math Olympiad, grade 8
[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$
[b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$.
[img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img]
[b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$.
[b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png
[/img]
[b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends.
[b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that
$$a_1 - 2a_2 + a_3 \le 0$$
$$a_2-2a_3 + a_ 4 \le 0$$
$$...$$
$$a_{98}-2a_{99 }+ a_{100} \le 0$$
and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].
1998 Tuymaada Olympiad, 3
The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.
2012 India Regional Mathematical Olympiad, 3
Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.
2004 Junior Balkan Team Selection Tests - Romania, 3
Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$.
Prove that the lines $OM$ and $AB$ are perpendicular.
2000 Rioplatense Mathematical Olympiad, Level 3, 5
Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$.
If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrents
2003 AIME Problems, 10
Two positive integers differ by $60.$ The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?
2006 China Team Selection Test, 1
Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \cdot BT \cdot BR = BF \cdot AT \cdot AR$.
2007 F = Ma, 23
If a planet of radius $R$ spins with an angular velocity $\omega$ about an axis through the North Pole, what is the ratio of the normal force experienced by a person at the equator to that experienced by a person at the North Pole? Assume a constant gravitational field $g$ and that both people are stationary relative to the planet and are at sea level.
$ \textbf{(A)}\ g/R\omega^2$
$\textbf{(B)}\ R\omega^2/g $
$\textbf{(C)}\ 1- R\omega^2/g$
$\textbf{(D)}\ 1+g/R\omega^2$
$\textbf{(E)}\ 1+R\omega^2/g $
2019 CHKMO, 1
Given that $a,b$, and $c$ are positive real numbers such that $ab + bc + ca \geq 1$, prove that
\[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \geq \frac{\sqrt{3}}{abc} .\]
2020 Online Math Open Problems, 30
Let $c$ be the smallest positive real number such that for all positive integers $n$ and all positive real numbers $x_1$, $\ldots$, $x_n$, the inequality \[ \sum_{k=0}^n \frac{(n^3+k^3-k^2n)^{3/2}}{\sqrt{x_1^2+\dots +x_k^2+x_{k+1}+\dots +x_n}} \leq \sqrt{3}\left(\sum_{i=1}^n \frac{i^3(4n-3i+100)}{x_i}\right)+cn^5+100n^4 \] holds. Compute $\lfloor 2020c \rfloor$.
[i]Proposed by Luke Robitaille[/i]
2020 LMT Fall, B22
A cube has one of its vertices and all edges connected to that vertex deleted. How many ways can the letters from the word "$AMONGUS$" be placed on the remaining vertices of the cube so that one can walk along the edges to spell out "$AMONGUS$"? Note that each vertex will have at most $1$ letter, and one vertex is deleted and not included in the walk
2014 PUMaC Number Theory A, 1
Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.
2007 Princeton University Math Competition, 2
Suppose that $A$ is a set of positive integers less than $N$ and that no two distinct elements of $A$ sum to a perfect square. That is, if $a_1, a_2 \in A$ and $a_1 \neq a_2$ then $|a_1+a_2|$ is not a square of an integer. Prove that the maximum number of elements in $A$ is at least $\left\lfloor\frac{11}{32}N\right\rfloor$ .
2001 Estonia Team Selection Test, 2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2021 Science ON all problems, 4
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]
2021 AMC 12/AHSME Spring, 22
Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$?
$\textbf{(A) }-\frac{3}{49} \qquad \textbf{(B) }-\frac{1}{28} \qquad \textbf{(C) }\frac{^3\sqrt7}{64} \qquad \textbf{(D) }\frac{1}{32}\qquad \textbf{(E) }\frac{1}{28}$
2023 Azerbaijan IMO TST, 4
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
2001 Nordic, 4
Let ${ABCDEF}$ be a convex hexagon, in which each of the diagonals ${AD, BE}$ , and ${CF}$ divides the hexagon into two quadrilaterals of equal area. Show that ${AD, BE}$ , and ${CF}$ are concurrent.
1969 IMO Longlists, 60
$(SWE 3)$ Find the natural number $n$ with the following properties:
$(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$
$(2)$ $n$ is the smallest integer with the above property.