Found problems: 85335
2008 ITAMO, 3
Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions:
(i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$;
(ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.
2014 Online Math Open Problems, 12
Let $a$, $b$, $c$ be positive real numbers for which \[
\frac{5}{a} = b+c, \quad
\frac{10}{b} = c+a, \quad \text{and} \quad
\frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i]
2004 South East Mathematical Olympiad, 2
In $\triangle$ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.
PEN Q Problems, 1
Suppose $p(x) \in \mathbb{Z}[x]$ and $P(a)P(b)=-(a-b)^2$ for some distinct $a, b \in \mathbb{Z}$. Prove that $P(a)+P(b)=0$.
2012 239 Open Mathematical Olympiad, 7
Vasya conceived a two-digit number $a$, and Petya is trying to guess it. To do this, he tells Vasya a natural number $k$, and Vasya tells Petya the sum of the digits of the number $ka$. What is the smallest number of questions that Petya has to ask so that he can certainly be able to determine Vasya’s number?
2014 Contests, 1
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
2019 Ramnicean Hope, 1
Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $
[i]Ovidiu Țâțan[/i]
2024 Belarus Team Selection Test, 3.4
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
[i]N. Sheshko, D. Zmiaikou[/i]
2015 AMC 10, 3
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?
$ \textbf{(A) }9\qquad\textbf{(B) }18\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24 $
[asy]
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defaultpen(linewidth(0.8));
path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1)
{
for(int j=0;j<=3-i;j=j+1)
{
filldraw(shift((i,j))*h,black);
filldraw(shift((j,i))*v,black);
}
}[/asy]
2021 China Team Selection Test, 1
Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.
2010 LMT, 2
Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$
2008 National Olympiad First Round, 13
Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D\in [AB]$ and $[DC]\perp [BC]$. If $m(\widehat{ABC})=\alpha$, $m(\widehat{BCA})=3\alpha$, and $|AC|-|AD|=10$, what is $|BD|$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 18
\qquad\textbf{(D)}\ 20
\qquad\textbf{(E)}\ 22
$
2022 VIASM Summer Challenge, Problem 2
Give $P(x) = {x^{2022}} + {a_{2021}}{x^{2021}} + ... + {a_1}x + 1$ is a polynomial with real coefficents.
a) Assume that $2021a_{2021}^2 - 4044{a_{2020}} < 0.$ Prove that: $P(x)$ can't have $2022$ real roots.
b) Assume that $a_1^2 + a_2^2 + ... + a_{2021}^2 \le \frac{4}{{2021}}.$ Prove that: $P(x)\ge 0$, for all $x\in \mathbb{R}.$
1967 Vietnam National Olympiad, 3
i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$.
ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon.
iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.
2013 JBMO Shortlist, 2
$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.
1995 AMC 12/AHSME, 7
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:
$\textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 75 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 100$
2024 Saint Petersburg Mathematical Olympiad, 6
Call a positive integer number $n$ [i]poor[/i] if equation \[x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) \] has no solutions in positive integers $1<x_i<n$. Does there exist poor number, which has more than $100 \ 000$ distinct prime divisors?
1907 Eotvos Mathematical Competition, 1
If $p$ and $q$ are odd integers, prove that the equation
$$x^2 + 2px + 2q = 0$$
has no rational roots.
2023 IFYM, Sozopol, 6
Alex and Bobby take turns playing the following game on an initially white row of $5000$ cells, with Alex starting first. On her turn, Alex must color two adjacent white cells black. On his turn, Bobby must color either one or three consecutive white cells black. No player can make a move after which there will be a white cell with no adjacent white cell. The game ends when one player cannot make a move (in which case that player loses), or when the entire row is colored black (in which case Alex wins). Who has a winning strategy?
2005 All-Russian Olympiad Regional Round, 11.7
11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$.
([i]A. Golovanov[/i])
2019 Brazil Team Selection Test, 2
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
2009 ELMO Problems, 6
Let $p$ be an odd prime and $x$ be an integer such that $p \mid x^3 - 1$ but $p \nmid x - 1$. Prove that \[ p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).\][i]John Berman[/i]
2024 Saint Petersburg Mathematical Olympiad, 1
The $100 \times 100$ table is filled with numbers from $1$ to $10 \ 000$ as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?
2009 China Northern MO, 7
Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ ,
For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ .
Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .
2023 ELMO Shortlist, C7
A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\).
Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed.
[i]Proposed by Linus Tang[/i]