Found problems: 85335
2013 Bosnia And Herzegovina - Regional Olympiad, 4
$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$
$b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$
1949 Miklós Schweitzer, 1
Let an infinite sequence of measurable sets be given on the interval $ (0,1)$ the measures of which are $ \geq \alpha>0$. Show that there exists a point of $ (0,1)$ which belongs to infinitely many terms of the sequence.
2022 Israel National Olympiad, P4
Find all triples $(a,b,c)$ of integers for which the equation
\[x^3-a^2x^2+b^2x-ab+3c=0\]
has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.
2006 District Olympiad, 4
a) Find two sets $X,Y$ such that $X\cap Y =\emptyset$, $X\cup Y = \mathbb Q^{\star}_{+}$ and $Y = \{a\cdot b \mid a,b \in X \}$.
b) Find two sets $U,V$ such that $U\cap V =\emptyset$, $U\cup V = \mathbb R$ and $V = \{x+y \mid x,y \in U \}$.
2002 Mediterranean Mathematics Olympiad, 2
Suppose $x, y, a$ are real numbers such that $x+y = x^3 +y^3 = x^5 +y^5 = a$. Find all possible values of $a.$
2021 AMC 12/AHSME Spring, 17
Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$[center][asy]unitsize(100);
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5);
draw(A--B--C--D--cycle, black);
draw(A--P, black);
draw(B--P, black);
draw(C--P, black);
draw(D--P, black);
label("$A$",A,(-1,0));
label("$B$",B,(1,0));
label("$C$",C,(1,-0));
label("$D$",D,(-1,0));
label("$2$",E,(0,0));
label("$3$",F,(0,0));
label("$4$",G,(0,0));
label("$5$",H,(0,0));
dot(A^^B^^C^^D^^P);
[/asy][/center]
$\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}$
2018 Nepal National Olympiad, 1a
[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$.
[color=red]NOTE: There is a high chance that this problems was copied.[/color]
2016 IMC, 1
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
2013 Tuymaada Olympiad, 6
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2023-IMOC, C1
There are $n$ cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all $n$ such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.
2017 Silk Road, 1
On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this?
(Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.)
(Bogdanov. I)
1998 Belarus Team Selection Test, 3
Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be [i]good [/i] if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime.
a) Determine whether $(s,t)$ is itself a good pair;
bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.
2021 Harvard-MIT Mathematics Tournament., 8
For each positive real number $\alpha$, define
$$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$
Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$,
$$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$
Determine, with proof, the smallest positive size of $S$.
2000 APMO, 1
Compute the sum: $\sum_{i=0}^{101} \frac{x_i^3}{1-3x_i+3x_i^2}$ for $x_i=\frac{i}{101}$.
2024 Ecuador NMO (OMEC), 2
Let $s(n)$ the sum of digits of $n$. Find the greatest 3-digits number $m$ such that $3s(m)=s(3m)$.
2013 JBMO Shortlist, 3
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.
2016 Purple Comet Problems, 6
Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553.
1977 Germany Team Selection Test, 4
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
2008 Gheorghe Vranceanu, 2
Show that there is a natural number $ n $ that satisfies the following inequalities:
$$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$
2005 National High School Mathematics League, 1
The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is
$\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$
2024 LMT Fall, 11
A Pokemon fan walks into a store. An employee tells them that there are $2$ Pikachus, $3$ Eevees, $4$ Snorlaxes, and $5$ Bulbasaurs remaining inside the gacha machine. Given that this fan cannot see what is inside the Poké Balls before opening them, find the least number of Poké Balls they must buy in order to be sure to get one Pikachu and one Snorlax.
2002 All-Russian Olympiad Regional Round, 9.5
Is it possible to arrange the numbers $1, 2, . . . , 60$ in that order, so that the sum of any two numbers between which there is one number, divisible by $2$, the sum of any two numbers between which there are two numbers divisible by $3$, . . . , the sum of any two numbers between which there is are there six numbers, divisible by $7$?
1984 IMO Shortlist, 1
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
2019 International Zhautykov OIympiad, 5
Natural number $n>1$ is given. Let $I$ be a set of integers that are relatively prime to $n$. Define the function $f:I=>N$. We call a function $k-periodic$ if for any $a,b$ , $f(a)=f(b)$ whenever $ k|a-b $. We know that $f$ is $n-periodic$. Prove that minimal period of $f$ divides all other periods.
Example: if $n=6$ and $f(1)=f(5)$ then minimal period is 1, if $f(1)$ is not equal to $f(5)$ then minimal period is 3.
2024 HMNT, 1
The integers from $1$ to $9$ are arranged in a $3\times3$ grid. The rows and columns of the grid correspond to $6$ three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the $6$ numbers.