Found problems: 85335
2014 CentroAmerican, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
1994 Italy TST, 3
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
2020 Balkan MO Shortlist, N5
Consider an integer $n\geq 2$ and an odd prime $p$. Let $U$ be the set of all positive integers $($strictly$)$ less than $p^n$ that are not divisible by $p$, and let $N$ be the number of elements of $U$. Does there exist permutation $a_1,a_2,\cdots a_N$ of the numbers in $U$ such that the sum $\sum_{k=1}^N a_ka_{k+1}$,where $a_{N+1}=a_1$, be divisible by $p^{n-1}$ but not by $p^n$?
$Alexander \ Ivanov \, Bulgaria$
2010 Contests, 2
On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.
2007 ITest, 47
Let $\{X_n\}$ and $\{Y_n\}$ be sequences defined as follows: \[X_0=Y_0=X_1=Y_1=1,\] \begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*} Let $k$ be the largest integer that satisfies all of the following conditions: [list=i][*] $|X_i-k|\leq 2007$, for some positive integer $i$;
[*] $|Y_j-k|\leq 2007$, for some positive integer $j$; and
[*] $k<10^{2007}.$[/list] Find the remainder when $k$ is divided by $2007$.
2008 F = Ma, 21
Consider a particle at rest which may decay into two (daughter) particles or into three (daughter) particles. Which of the following is true in the two-body case but false in the three-body case? (There are no external forces.)
(a) The velocity vectors of the daughter particles must lie in a single plane.
(b) Given the total kinetic energy of the system and the mass of each daughter particle, it is possible to determine the speed of each daughter particle.
(c) Given the speed(s) of all but one daughter particle, it is possible to determine the speed of the remaining particle.
(d) The total momentum of the daughter particles is zero.
(e) None of the above.
2018 BMT Spring, 9
Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \,\,\, (mod \,\,\,199)$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$.
2021 LMT Spring, A30
Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Aditya Rao[/i]
2007 Harvard-MIT Mathematics Tournament, 15
Points $A$, $B$, and $C$ lie in that order on line $\ell$ such that $AB=3$ and $BC=2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.
1990 IMO Longlists, 34
There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.
1990 Tournament Of Towns, (247) 1
Find the maximum number of parts into which the $Oxy$-plane can be divided by $100$ graphs of different quadratic functions of the form $y = ax^2 + bx + c$.
(N.B. Vasiliev, Moscow)
2023 Durer Math Competition Finals, 16
What is the remainder of $2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))$ when it is divided by $2023$?
Here $\wedge$ is the exponential symbol, for example $2\wedge (3\wedge 2) = 2\wedge 9 = 512$. The power tower contains the integers from $2025$ to $1$ exactly once, except that the number $2023$ is missing.
2017 Sharygin Geometry Olympiad, P22
Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.
2024 All-Russian Olympiad, 7
Let $x_1$ and $x_2$ be positive integers. On a straight line, $y_1$ white segments and $y_2$ black segments are given, with $y_1 \ge x_1$ and $y_2 \ge x_2$. Suppose that no two segments of the same colour intersect (and do not have common ends). Moreover, suppose that for any choice of $x_1$ white segments and $x_2$ black segments, some pair of selected segments will intersect. Prove that $(y_1-x_1)(y_2-x_2)<x_1x_2$.
[i]Proposed by G. Chelnokov[/i]
1974 Canada National Olympiad, 6
An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say $n$, of postage which is unattainable while all amounts larger than $n$ are attainable? (Justify your answer.)
1992 Irish Math Olympiad, 3
Let $A$ be a nonempty set with $n$ elements. Find the number of ways of choosing a pair of subsets $(B,C)$ of $A$ such that $B$ is a nonempty subset of $C$.
2019 India Regional Mathematical Olympiad, 2
Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.
2009 Purple Comet Problems, 2
Let $p_1 = 2, p_2 = 3, p_3 = 5 ...$ be the sequence of prime numbers. Find the least positive even integer $n$ so that $p_1 + p_2 + p_3 + ... + p_n$ is not prime.
1996 Romania Team Selection Test, 14
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent:
(i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $;
(ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.
1994 Vietnam National Olympiad, 3
Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
2019 Junior Balkan Team Selection Tests - Moldova, 6
Let $p$ and $q$ be integers. If $k^2+pk+q>0$ for every integer $k$, show that $x^2+px+q>0$ for every real number $x$.
2024 Kazakhstan National Olympiad, 4
Players $A$ and $B$ play the following game on the coordinate plane. Player $A$ hides a nut at one of the points with integer coordinates, and player $B$ tries to find this hidden nut. In one move $B$ can choose three different points with integer coordinates, then $A$ tells whether these three points together with the nut's point lie on the same circle or not. Can $B$ be guaranteed to find the nut in a finite number of moves?
2025 International Zhautykov Olympiad, 4
Vaysha has a board with $999$ consecutive numbers written and $999$ labels of the form [i]"This number is [b]not [/b]divisible by $i$"[/i], for $i \in \{ 2,3, \dots ,1000 \} $. She places each label next to a number on the board, so that each number has exactly one label. For each true statement on the stickers, Vaysha gets a piece of candy. How many pieces of candy can Vaysha guarantee to win, regardless of the numbers written on the board, if she plays optimally?
2003 AIME Problems, 11
An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.
1971 All Soviet Union Mathematical Olympiad, 151
Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.