Found problems: 85335
2000 All-Russian Olympiad Regional Round, 10.5
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality
$$|f(x + y) + \sin x + \sin y| < 2?$$
1989 IMO Shortlist, 1
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
2016 China Girls Math Olympiad, 5
Define a sequence $\{a_n\}$ by\[S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).\]
Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$.
For any positive integer $n$ ,prove that\[a_{n}\ge \frac{4}{\sqrt{9n+7}}.\]
2013 China Team Selection Test, 2
Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying:
$(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $;
$(2)$ For any positive integer $n$, $a_n<1.01^n K$;
$(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.
2000 Tournament Of Towns, 2
What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon?
(A Shapovalov)
2013 Iran MO (3rd Round), 2
Real numbers $a_1 , a_2 , \dots, a_n$ add up to zero. Find the maximum of $a_1 x_1 + a_2 x_2 + \dots + a_n x_n$ in term of $a_i$'s, when $x_i$'s vary in real numbers such that $(x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_{n-1} - x_n)^2 \leq 1$.
(15 points)
1995 Czech and Slovak Match, 6
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
2019 Israel National Olympiad, 2
We are given a 5x5 square grid, divided to 1x1 tiles. Two tiles are called [b]linked[/b] if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture the gray tiles are the ones linked to the red tile.
[img]https://i.imgur.com/JVTQ9wB.png[/img]
Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?
2004 VTRMC, Problem 5
Let $f(x)=\int^x_0\sin(t^2-t+x)dt$. Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$.
1996 Argentina National Olympiad, 6
In a tennis tournament of $10$ players, everyone played against everyone once. In this tournament, if player $i$ won the match against player $j$, then the total number of matches $i$ lost plus the total number of matches $j$ won is greater than or equal to $8$. We will say that three players $i$, $j$, $k$ form an [i]atypical tri[/i]o if $i$ beat $j$, $j$ beat $k$ and $k$ beat $i$. Prove that in the tournament there were exactly $40$ atypical trios.
2006 Germany Team Selection Test, 1
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
2022 Kyiv City MO Round 2, Problem 1
a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$?
b) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$?
Here $[a, b]$ denotes the least common multiple of integers $a, b$.
III Soros Olympiad 1996 - 97 (Russia), 9.5
An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.
2012 IMO, 1
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$
(The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)
[i]Proposed by Evangelos Psychas, Greece[/i]
2022 Junior Balkan Team Selection Tests - Romania, P4
Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$
1982 Bulgaria National Olympiad, Problem 1
Find all pairs of natural numbers $(n,k)$ for which
$(n+1)^{k}-1 = n!$.
2020 Korea National Olympiad, 1
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
2010 APMO, 1
Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.
2013 Online Math Open Problems, 13
In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots \\
1 & 2 & 3 & 4 & 5 & \cdots \\
1 & 3 & 6 & 10 & 15 & \cdots \\
1 & 4 & 10 & 20 & 35 & \cdots \\
1 & 5 & 15 & 35 & 70 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
\]
Wanda the Worm, who is on a diet after a feast two years ago, wants to eat $n$ numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of $n$?
[i]Proposed by Evan Chen[/i]
2009 Bosnia and Herzegovina Junior BMO TST, 3
Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$
2021 Ukraine National Mathematical Olympiad, 8
There are $101$ not necessarily different weights, each of which weighs an integer number of grams from $1$ g to $2020$ g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than $2020$. What is the largest number of grams can weigh all $101$ weights?
(Bogdan Rublev)
2021 Science ON grade XI, 3
$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix}
a &b &x\\
f(a) &f(b) &f(x)\\
g(a) &g(b) &g(x)\\
\end{vmatrix}$$
Prove that $h$ is differentiable and find $h'$.
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$\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $.
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[i](Sergiu Novac)[/i]
2022 IMC, 6
Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.
2024 Brazil National Olympiad, 2
A partition of a set \( A \) is a family of non-empty subsets of \( A \), such that any two distinct subsets in the family are disjoint, and the union of all subsets equals \( A \). We say that a partition of a set of integers \( B \) is [i]separated[/i] if each subset in the partition does [b]not[/b] contain consecutive integers. Prove that, for every positive integer \( n \), the number of partitions of the set \( \{1, 2, \dots, n\} \) is equal to the number of separated partitions of the set \( \{1, 2, \dots, n+1\} \).
For example, \( \{\{1,3\}, \{2\}\} \) is a separated partition of the set \( \{1,2,3\} \). On the other hand, \( \{\{1,2\}, \{3\}\} \) is a partition of the same set, but it is not separated since \( \{1,2\} \) contains consecutive integers.
2025 Kosovo EGMO Team Selection Test, P3
The numbers 1, 2, ... , 36 are written in the cells of a $6 \times 6$ grid. Two cells are called neighbors if they have a common side or vertex. A frog is located at the cell with the number 1 written on it. Every minute, if a neighboring cell has a bigger number than the cell where the frog is located, the frog jumps to the neighboring cell that has the biggest number written on it. The frog continues like this until there are no neighboring cells with a bigger number than the cell where the frog is located. What is the biggest possible number of jumps the frog can make?