Found problems: 85335
2016 Junior Balkan MO, 2
Let $a,b,c $be positive real numbers.Prove that
$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.
2012 Online Math Open Problems, 6
Alice's favorite number has the following properties:
[list]
[*] It has 8 distinct digits.
[*]The digits are decreasing when read from left to right.
[*]It is divisible by 180.[/list]
What is Alice's favorite number?
[i]Author: Anderson Wang[/i]
2004 All-Russian Olympiad, 2
Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.
2021 AMC 12/AHSME Fall, 13
Let $c = \frac{2\pi}{11}.$ What is the value of
$$\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$$
$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{\sqrt{-11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\
\frac{10}{11} \qquad\textbf{(E)}\ 1$
2016 Croatia Team Selection Test, Problem 2
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.
Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
1979 IMO Longlists, 81
Let $\Pi$ be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped $P_0$ can be decomposed into parallelepipeds $P_1,P_2, . . . ,P_N\in \Pi$, prove that $P_0\in \Pi$.
2020 Iranian Combinatorics Olympiad, 3
$1399$ points and some chords between them is given.
$a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]exactly one[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.
$b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]both[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.
[i]Proposed by Afrouz Jabalameli, Abolfazl Asadi[/i]
2015 Silk Road, 2
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
2021 Belarusian National Olympiad, 10.4
Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots.
a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly.
b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$
1974 IMO Longlists, 52
A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that:
(a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves.
(b) If $2u\le v$, the rabbit can always run away from the fox.
2013 Stanford Mathematics Tournament, 10
Consider a sequence given by $a_n=a_{n-1}+3a_{n-2}+a_{n-3}$, where $a_0=a_1=a_2=1$. What is the remainder of $a_{2013}$ divided by $7$?
2002 Czech-Polish-Slovak Match, 2
A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.
2024 CMIMC Combinatorics and Computer Science, 10
Suppose 100 people are gathered around at a park, each with an envelope with their name on it (all their names are distinct). Then, the envelopes are uniformly and randomly permuted between the people. If $N$ is the number of people who end up with their original envelope, find the expected value of $N^5$.
[i]Proposed by Michael Duncan[/i]
2019 Belarus Team Selection Test, 8.2
Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions:
1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$;
2. $f$ takes all integer values.
[i](I. Voronovich)[/i]
1995 Kurschak Competition, 1
Given in the plane is a lattice and a grid rectangle with sides parallel to the coordinate axes. We divide the rectangle into grid triangles with area $\frac12$. Prove that the number of right angled triangles is at least twice as much as the shorter side of the rectangle.
(A grid polygon is a polygon such that both coordinates of each vertex is an integer.)
2019 India PRMO, 12
A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers
$$a_1 < a_2 < \cdots < a_k$$
such that
$$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$.
Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.
1991 All Soviet Union Mathematical Olympiad, 548
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
2020 Korea National Olympiad, 1
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
2014 India Regional Mathematical Olympiad, 2
Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let
$(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$
$(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$
$(iii)$ $a_n+a_{n+1}=z$
Express $d$ in terms of $x,y,z,n$
1983 Bulgaria National Olympiad, Problem 3
A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively.
(a) Prove that $AP/AD=BQ/BC$.
(b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.
2001 AMC 10, 22
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y \plus{} z$.
$ \textbf{(A)}\ 43 \qquad
\textbf{(B)}\ 44 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 46 \qquad
\textbf{(E)}\ 47$
[asy]unitsize(10mm);
defaultpen(linewidth(1pt));
for(int i=0; i<=3; ++i)
{
draw((0,i)--(3,i));
draw((i,0)--(i,3));
}
label("$25$",(0.5,0.5));
label("$z$",(1.5,0.5));
label("$21$",(2.5,0.5));
label("$18$",(0.5,1.5));
label("$x$",(1.5,1.5));
label("$y$",(2.5,1.5));
label("$v$",(0.5,2.5));
label("$24$",(1.5,2.5));
label("$w$",(2.5,2.5));[/asy]
1959 AMC 12/AHSME, 29
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $
1952 AMC 12/AHSME, 25
A powderman set a fuse for a blast to take place in $ 30$ seconds. He ran away at a rate of $ 8$ yards per second. Sound travels at the rate of $ 1080$ feet per second. When the powderman heard the blast, he had run approximately:
$ \textbf{(A)}\ 200 \text{ yd.} \qquad\textbf{(B)}\ 352 \text{ yd.} \qquad\textbf{(C)}\ 300 \text{ yd.} \qquad\textbf{(D)}\ 245 \text{ yd.} \qquad\textbf{(E)}\ 512 \text{ yd.}$
2018 Iran Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions:
a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$
b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval.
[i]Proposed by Navid Safaei[/i]
2019 Purple Comet Problems, 3
The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length $36\pi$. Find $n$ such that the area of the shaded region is $n\pi$.
[img]https://cdn.artofproblemsolving.com/attachments/4/5/c9ffdc41c633cc61127ef585a45ee5e6c0f88d.png[/img]