Found problems: 85335
2024 JHMT HS, 5
Compute the positive difference between the two solutions to the equation $2x^2-28x+9=0$.
2010 Malaysia National Olympiad, 6
Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]
2011 Uzbekistan National Olympiad, 3
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$
2001 Bundeswettbewerb Mathematik, 1
On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process?
a.) $ T \equal{} 1000$ (Cono Sur)
b.) $ T \equal{} 2001$ (BWM)
1987 Tournament Of Towns, (146) 3
In a certain city only simple (pairwise) exchanges of apartments are allowed (if two families exchange fiats , they are not allowed to participate in another exchange on the same day). Prove that any compound exchange may be effected in two days. It is assumed that under any exchange (simple or comp ound) each family occupies one fiat before and after the exchange and the family cannot split up .
(A . Shnirelman , N .N . Konstantinov)
2022 Olimphíada, 2
We say that a real $a\geq-1$ is philosophical if there exists a sequence $\epsilon_1,\epsilon_2,\dots$, with $\epsilon_i \in\{-1,1\}$ for all $i\geq1$, such that the sequence $a_1,a_2,a_3,\dots$, with $a_1=a$, satisfies
$$a_{n+1}=\epsilon_{n}\sqrt{a_{n}+1},\forall n\geq1$$
and is periodic. Find all philosophical numbers.
2017 CCA Math Bonanza, L1.4
Wild Bill goes to Las Vejas and takes part in a special lottery called [i]Reverse Yrettol[/i]. In this lottery, a player may buy a ticket on which he or she may select $5$ distinct numbers from $1-20$ (inclusive). Then, $5$ distinct numbers from $1-20$ are drawn at random. A player wins if his or her ticket contains [i]none[/i] of the numbers which were drawn. If Wild Bill buys a ticket, what is the probability that he will win?
[i]2017 CCA Math Bonanza Lightning Round #1.4[/i]
2022 Kyiv City MO Round 2, Problem 3
Nonzero real numbers $x_1, x_2, \ldots, x_n$ satisfy the following condition:
$$x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}$$
Determine all $n$ for which $x_1, x_2, \ldots, x_n$ have to be equal.
[i](Proposed by Oleksii Masalitin, Anton Trygub)[/i]
2003 Baltic Way, 20
Suppose that the sum of all positive divisors of a natural number $n$, $n$ excluded, plus the number of these divisors is equal to $n$. Prove that $n = 2m^2$ for some integer $m$.
2021 Alibaba Global Math Competition, 2
The winners of first AGMC in 2019 gifts the person in charge of the organiser, which is a polyhedron formed by $60$ congruent triangles. From the photo, we can see that this polyhedron formed by $60$ quadrilateral spaces.
(Note: You can find the photo in 3.4 of [url]https://files.alicdn.com/tpsservice/18c5c7b31a7074edc58abb48175ae4c3.pdf?spm=a1zmmc.index.0.0.51c0719dNAbw3C&file=18c5c7b31a7074edc58abb48175ae4c3.pdf[/url])
A quadrilateral space is the plane figures that we fold the figures following the diagonal on a $n$ sides on a plane (i.e. form an appropriate dihedral angle in where the chosen diagonal is). "Two figure spaces are congruent" means they can coincide completely by isometric transform in $\mathbb{R}^3$. A polyhedron is the bounded space region, whose boundary is formed by the common edge of finite polygon.
(a) We know that $2021=43\times 47$. Does there exist a polyhedron, whose surface can be formed by $43$ congruent $47$-gon?
(b) Prove your answer in (a) with logical explanation.
2011 Czech and Slovak Olympiad III A, 4
Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.
2009 Miklós Schweitzer, 1
On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?
2024 Korea National Olympiad, 4
Find the smallest positive integer \( k \geq 2 \) for which there exists a polynomial \( f(x) \) of degree \( k \) with integer coefficients and a leading coefficient of \( 1 \) that satisfies the following condition:
(Condition) For any two integers \( m \) and \( n \), if \( f(m) - f(n) \) is a multiple of \( 31 \), then \( m - n \) is a multiple of \( 31 \).
2017 Bulgaria National Olympiad, 1
An convex qudrilateral $ABCD$ is given. $O$ is the intersection point of the diagonals $AC$ and $BD$. The points $A_1,B_1,C_1, D_1$ lie respectively on $AO, BO, CO, DO$ such that $AA_1=CC_1, BB_1=DD_1$.
The circumcircles of $\triangle AOB$ and $\triangle COD$ meet at second time at $M$ and the the circumcircles of $\triangle AOD$ and $\triangle BOC$ - at $N$.
The circumcircles of $\triangle A_1OB_1$ and $\triangle C_1OD_1$ meet at second time at $P$ and the the circumcircles of $\triangle A_1OD_1$ and $\triangle B_1OC_1$ - at $Q$.
Prove that the quadrilateral $MNPQ$ is cyclic.
2017 Mathematical Talent Reward Programme, MCQ: P 5
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!\cdot b!\cdot c!\cdot d!=24!$$
[list=1]
[*] 4
[*] 4!
[*] $4^4$
[*] None of these
[/list]
2019 AMC 8, 17
What is the value of the product $$\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?$$
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) } 50$
2006 USA Team Selection Test, 1
A communications network consisting of some terminals is called a [i]$3$-connector[/i] if among any three terminals, some two of them can directly communicate with each other. A communications network contains a [i]windmill[/i] with $n$ blades if there exist $n$ pairs of terminals $\{x_{1},y_{1}\},\{x_{2},y_{2}\},\ldots,\{x_{n},y_{n}\}$ such that each $x_{i}$ can directly communicate with the corresponding $y_{i}$ and there is a [i]hub[/i] terminal that can directly communicate with each of the $2n$ terminals $x_{1}, y_{1},\ldots,x_{n}, y_{n}$ . Determine the minimum value of $f (n)$, in terms of $n$, such that a $3$ -connector with $f (n)$ terminals always contains a windmill with $n$ blades.
1998 China Team Selection Test, 1
In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.
2014 AIME Problems, 3
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.
2019 Kurschak Competition, 2
Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.
2008 Korea Junior Math Olympiad, 3
For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.
2015 China Team Selection Test, 4
Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals.
Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.
2012 IFYM, Sozopol, 7
The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that
$(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$
and $(a-c)^2\leq 2b^2\leq (a+c)^2$.
STEMS 2021 Math Cat A, Q5
Let $ABC$ be a triangle with $I$ as incenter.The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle.Make a tangent at $D'$ to incircle.Let it meet $(ABC)$ at $X,Y$ respectively.Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$.Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$
2019 IMEO, 3
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all real $x, y$, the following relation holds: $$(x+y) \cdot f(x+y)= f(f(x)+y) \cdot f(x+f(y)).$$
[i]Proposed by Vadym Koval (Ukraine)[/i]