Found problems: 85335
1998 Singapore Team Selection Test, 1
The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that:
\[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}.
\]
1993 IMO Shortlist, 4
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$:
$\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$
III Soros Olympiad 1996 - 97 (Russia), 11.9
Given a regular hexagon with a side of $100$. Each side is divided into one hundred equal parts. Through the division points and vertices of the hexagon, all sorts of straight lines parallel to its sides are drawn. These lines divided the hexagon into single regular triangles. Consider covering a hexagon with equal rhombuses. Each rhombus is made up of two triangles. (These rhombuses cover the entire hexagon and do not overlap.) Among the lines that form our grid, we select those that intersect exactly to the rhombuses (intersect diagonally). How many such lines will there be if:
a) $k = 101$;
b) $k = 100$;
c) $k = 87$?
2010 AMC 12/AHSME, 14
Let $ a$, $ b$, $ c$, $ d$, and $ e$ be positive integers with $ a\plus{}b\plus{}c\plus{}d\plus{}e\equal{}2010$, and let $ M$ be the largest of the sums $ a\plus{}b$, $ b\plus{}c$, $ c\plus{}d$, and $ d\plus{}e$. What is the smallest possible value of $ M$?
$ \textbf{(A)}\ 670 \qquad
\textbf{(B)}\ 671 \qquad
\textbf{(C)}\ 802 \qquad
\textbf{(D)}\ 803 \qquad
\textbf{(E)}\ 804$
2011 Dutch BxMO TST, 3
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
2006 Estonia National Olympiad, 2
Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.
2020 Iran Team Selection Test, 3
Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.
[i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]
2004 Tournament Of Towns, 3
Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty.
We can perform any combination of the following operations:
- Pour away the entire amount in bucket $X$,
- Pour the entire amount in bucket $X$ into bucket $Y$,
- Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount.
[b](a)[/b] How can we obtain 10 litres of 30% syrup if $n = 20$?
[b](b)[/b] Determine all possible values of $n$ for which the task in (a) is possible.
KoMaL A Problems 2017/2018, A. 709
Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle
C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.
2016 CMIMC, 8
Consider the sequence of sets defined by $S_0=\{0,1\},S_1=\{0,1,2\}$, and for $n\ge2$, \[S_n=S_{n-1}\cup\{2^n+x\mid x\in S_{n-2}\}.\] For example, $S_2=\{0,1,2\}\cup\{2^2+0,2^2+1\}=\{0,1,2,4,5\}$. Find the $200$th smallest element of $S_{2016}$.
1970 IMO Longlists, 18
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
1998 Croatia National Olympiad, Problem 1
Which number is greater:
$$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?
2023 IFYM, Sozopol, 1
On the board, the numbers from $1$ to $n$ are written. Achka (A) and Bavachka (B) play the following game. First, A erases one number, then B erases two consecutive numbers, then A erases three consecutive numbers, and finally B erases four consecutive numbers. What is the smallest $n$ such that B can definitely make her moves, no matter how A plays?
2023 LMT Fall, 2
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{7}$
Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games.
[/hide]
2023 OMpD, 3
Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements.
[b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.
2019 MMATHS, 1
$S$ is a set of positive integers with the following properties:
(a) There are exactly $3$ positive integers missing from $S$.
(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow a and b to be the same.)
Find all possibilities for the set $S$ (with proof).
2017 AIME Problems, 10
Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i, $ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
1990 Tournament Of Towns, (259) 3
A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case.
(D. Fomin, Leningrad)
2007 Estonia Math Open Junior Contests, 2
The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.
2000 Croatia National Olympiad, Problem 4
Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that
$$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$
1994 IMO Shortlist, 3
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
2013 Moldova Team Selection Test, 1
For any positive real numbers $x,y,z$, prove that
$\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$
2017 Puerto Rico Team Selection Test, 4
Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is:
i) $3 \times 3$
ii) $3 \times 4$
2017 ASDAN Math Tournament, 6
You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?
1990 Greece National Olympiad, 4
Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$