Found problems: 85335
2006 AMC 8, 23
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 5$
2021 CCA Math Bonanza, L1.1
Compute
\[
(2+0\cdot 2 \cdot 1)+(2+0-2) \cdot (1) + (2+0)\cdot (2-1) + (2) \cdot \left(0+2^{-1}\right).
\]
[i]2021 CCA Math Bonanza Lightning Round #1.1[/i]
2003 National Olympiad First Round, 25
Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$?
$
\textbf{(A)}\ 56\sqrt 3
\qquad\textbf{(B)}\ 56 \sqrt 2
\qquad\textbf{(C)}\ 50 \sqrt 2
\qquad\textbf{(D)}\ 84
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2019 Thailand Mathematical Olympiad, 9
A [i]chaisri[/i] figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon.
A [i]thubkaew[/i] figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.
Determine the maximum number of vertices that a thubkaew figure may have.
KoMaL A Problems 2019/2020, A. 779
Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$
Prove that lines $PK$ are concurrent.
PEN S Problems, 14
Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \le y$ and $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is nonnegative and as small as possible.
2015 Hanoi Open Mathematics Competitions, 11
Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.
2024 LMT Fall, 13
Some math team members decide to study at Cary Library after school. They walk $6$ blocks north, then $8$ blocks west to get there. If they walk $n$ blocks east from the library, they can buy boba from CoCo's. If CoCo's is the same distance from school as it is from the library, find $n$.
2005 Germany Team Selection Test, 1
Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.
1976 AMC 12/AHSME, 7
If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if
$\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$
$\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$
2006 MOP Homework, 6
Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial
$P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.
2013 NIMO Problems, 4
Consider a set of $1001$ points in the plane, no three collinear. Compute the minimum number of segments that must be drawn so that among any four points, we can find a triangle.
[i]Proposed by Ahaan S. Rungta / Amir Hossein[/i]
1992 Bundeswettbewerb Mathematik, 4
For three sequences $(x_n),(y_n),(z_n)$ with positive starting elements $x_1,y_1,z_1$ we have the following formulae:
\[ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} \quad z_{n+1} = x_n + \frac{1}{y_n} \quad (n = 1,2,3, \ldots)\]
a.) Prove that none of the three sequences is bounded from above.
b.) At least one of the numbers $x_{200},y_{200},z_{200}$ is greater than 20.
2010 Hanoi Open Mathematics Competitions, 9
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?
2022 Durer Math Competition Finals, 15
An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability $\frac13$ , left with probability $\frac13$ , and right with probability $\frac13$. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point?
If the answer is $\frac{p}{q}$ , where $p$ and $q$ are coprime positive integers, then your answer should be $p + q$.
[i]The steps of the ant are independent.[/i]
2018 Bosnia And Herzegovina - Regional Olympiad, 1
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions
1998 Gauss, 16
Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in
the diagram. If the two digit number is subtracted from the
three digit number, what is the smallest difference?
$\textbf{(A)}\ 269 \qquad \textbf{(B)}\ 278 \qquad \textbf{(C)}\ 484 \qquad \textbf{(D)}\ 271 \qquad \textbf{(E)}\ 261$
2025 CMIMC Combo/CS, 10
Let $a_n$ be the number of ways to express $n$ as an ordered sum of powers of $3.$ For example $a_4=3,$ since $$4=1+1+1+1=1+3=3+1.$$ Let $b_n$ denote the remainder upon dividing $a_n$ by $3.$ Evaluate $$\sum_{n=1}^{3^{2025}} b_n.$$
2006 Cuba MO, 4
Let $f : Z_+ \to Z_+$ such that:
a) $f(n + 1) > f(n)$ for all $n \in Z_+$
b) $f(n + f(m)) = f(n) + m + 1$ for all $n,m \in Z_+$
Find $f(2006)$.
1981 Dutch Mathematical Olympiad, 3
We want to split the set of natural numbers from $1$ to $3n$, where $n$ is a natural number, into $n$ mutually disjoint sets $\{x,y,z\}$ of three elements such that always holds: $x + y = 3z$. Is this possible for :
a) $n = 5$?
b) $n=10$?
In both cases, provide either such a split or proof that such a split is not possible.
2001 USA Team Selection Test, 3
For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that
(i) $B \subseteq A$;
(ii) $|B| \ge 668$;
(iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$.
2021 Princeton University Math Competition, 14
Heron is going to watch a show with $n$ episodes which are released one each day. Heron wants to watch the first and last episodes on the days they first air, and he doesn’t want to have two days in a row that he watches no episodes. He can watch as many episodes as he wants in a day. Denote by $f(n)$ the number of ways Heron can choose how many episodes he watches each day satisfying these constraints. Let $N$ be the $2021$st smallest value of $n$ where $f(n) \equiv 2$ mod $3$. Find $N$.
2017 Switzerland - Final Round, 3
The main building of ETH Zurich is a rectangle divided into unit squares. Every side of a square is a wall, with certain walls having doors. The outer wall of the main building has no doors. A number of participants of the SMO have gathered in the main building lost. You can only move from one square to another through doors. We have indicates that there is a walkable path between every two squares of the main building.
Cyril wants the participants to find each other again by having everyone on the same square leads. To do this, he can give them the following instructions via walkie-talkie: North, East, South or West. After each instruction, each participant simultaneously attempts a square in that direction to go. If there is no door in the corresponding wall, he remains standing.
Show that Cyril can reach his goal after a finite number of directions, no matter which one square the participants at the beginning.
[hide=original wording]Das Hauptgebäude der ETH Zürich ist ein in Einheitsquadrate unterteiltes Rechteck. Jede Seite eines Quadrates ist eine Wand, wobei gewisse Wände Türen haben. Die Aussenwand des Hauptgebäudes hat keine Türen. Eine Anzahl von Teilnehmern der SMO hat sich im Hauptgebäude verirrt. Sie können sich nur durch Türen von einem Quadrat zum anderen bewegen. Wir nehmen an, dass zwischen je zwei Quadraten des Hauptgebäudes ein begehbarer Weg existiert.
Cyril möchte erreichen, dass sich die Teilnehmer wieder nden, indem er alle auf dasselbe Quadrat führt. Dazu kann er ihnen per Walkie-Talkie folgende Anweisungen geben: Nord, Ost, Süd oder West. Nach jeder Anweisung versucht jeder Teilnehmer gleichzeitig, ein Quadrat in diese Richtung zu gehen. Falls in der entsprechenden Wand keine Türe ist, bleibt er stehen.
Zeige, dass Cyril sein Ziel nach endlich vielen Anweisungen erreichen kann, egal auf welchen Quadraten sich die Teilnehmer am Anfang benden. [/hide]
2005 Belarusian National Olympiad, 1
Prove for positive numbers:
$$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$
2014 Belarus Team Selection Test, 2
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$