Found problems: 85335
2021 Benelux, 1
(a) Prove that for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$,
\[
\max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0.
\]
(b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$.
1978 IMO Longlists, 21
A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.
1982 Bulgaria National Olympiad, Problem 5
Find all values of parameters $a,b$ for which the polynomial
$$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.
2019 EGMO, 6
On a circle, Alina draws $2019$ chords, the endpoints of which are all different. A point is considered [i]marked[/i] if it is either
$\text{(i)}$ one of the $4038$ endpoints of a chord; or
$\text{(ii)}$ an intersection point of at least two chords.
Alina labels each marked point. Of the $4038$ points meeting criterion $\text{(i)}$, Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$. She labels each point meeting criterion $\text{(ii)}$ with an arbitrary integer (not necessarily positive).
Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with $k$ marked points has $k-1$ such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference.
Alina finds that the $N + 1$ yellow labels take each value $0, 1, . . . , N$ exactly once. Show that at least one blue label is a multiple of $3$.
(A chord is a line segment joining two different points on a circle.)
1999 USAMTS Problems, 2
Let $C$ be the set of non-negative integers which can be expressed as $1999s+2000t$, where $s$ and $t$ are also non-negative integers.
(a) Show that $3,994,001$ is not in $C$.
(b) Show that if $0\leq n \leq 3,994,001$ and $n$ is an integer not in $C$, then $3,994,001-n$ is in $C$.
1987 Putnam, B1
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]
2013 Swedish Mathematical Competition, 3
Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$
2012 Online Math Open Problems, 18
The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem should ask for the positive difference.[/hide]
1997 Greece Junior Math Olympiad, 4
Consider ten concentric circles and ten rays as in the following figure.
At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as:
a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also
b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''.
Prove that the sum of the $100$ signed numbers that occur, equals zero.
[img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]
2022/2023 Tournament of Towns, P6
Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$?
[i]Viktor Kleptsyn[/i]
2017 Princeton University Math Competition, A2
Let $a_1, a_2, a_3, ...$ be a monotonically decreasing sequence of positive real numbers converging to zero. Suppose that $\Sigma_{i=1}^{\infty}\frac{a_i}{i}$ diverges. Show that $\Sigma_{i=1}^{\infty}a_i^{2^{2017}}$ also diverges. You may assume in your proof that $\Sigma_{i=1}^{\infty}\frac{1}{i^p}$ converges for all real numbers $p > 1$. (A sum $\Sigma_{i=1}^{\infty}b_i$ of positive real numbers $b_i$ diverges if for each real number $N$ there is a positive integer $k$ such that $b_1+b_2+...+b_k > N$.)
2009 Greece Team Selection Test, 1
Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.
2005 Taiwan National Olympiad, 1
Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]
1997 Slovenia National Olympiad, Problem 1
Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.
2010 Kazakhstan National Olympiad, 2
Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$.
Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ .
Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.
2001 AIME Problems, 3
Given that
\begin{align*}
x_{1}&=211,\\
x_{2}&=375,\\
x_{3}&=420,\\
x_{4}&=523, \text{ and}\\
x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \text{ when } n \geq 5,
\end{align*}
find the value of $x_{531}+x_{753}+x_{975}$.
2024 Indonesia TST, 1
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
[/list]
2017 Iberoamerican, 1
For every positive integer $n$ let $S(n)$ be the sum of its digits. We say $n$ has a property $P$ if all terms in the infinite secuence $n, S(n), S(S(n)),...$ are even numbers, and we say $n$ has a property $I$ if all terms in this secuence are odd. Show that for, $1 \le n \le 2017$ there are more $n$ that have property $I$ than those who have $P$.
2015 CCA Math Bonanza, L3.2
In triangle $ABC$, points $M$, $N$, and $P$ lie on sides $\overline{AC}$, $\overline{AB}$, and $\overline{BC}$, respectively. If $\angle{ABC}=42^\circ$, $\angle{MAN}=91^\circ$, and $\angle{NMA}=47^\circ$, compute $\frac{CB}{BP}$.
[i]2015 CCA Math Bonanza Lightning Round #3.2[/i]
2013 India Regional Mathematical Olympiad, 3
Consider the expression \[2013^2+2014^2+2015^2+ \cdots+n^2\]
Prove that there exists a natural number $n > 2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal $9999$
2012 JHMT, 10
A large flat plate of glass is suspended $\sqrt{2/3}$ units above a large flat plate of wood. (The glass is infinitely thin and causes no funny refractive effects.) A point source of light is suspended $\sqrt{6}$ units above the glass plate. An object rests on the glass plate of the following description. Its base is an isosceles trapezoid $ABCD$ with $AB \parallel DC$, $AB = AD = BC = 1$, and $DC = 2$. The point source of light is directly above the midpoint of $CD$. The object’s upper face is a triangle $EF G$ with $EF = 2$, $EG = F G =\sqrt3$. $G$ and $AB$ lie on opposite sides of the rectangle $EF CD$. The other sides of the object are $EA = ED = 1$, $F B = F C = 1$, and $GD = GC = 2$. Compute the area of the shadow that the object casts on the wood plate.
2018 China Second Round Olympiad, 2
In triangle $\triangle ABC$, $AB<AC$, $M,D,E$ are the midpoints of $BC$, the arcs $BAC$ and $BC$ of the circumcircle of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $AB$ at $F$, $AE$ meets $BC$ at $G$, and the perpendicular to $AB$ at $B$ meets segment $EF$ at $N$. If $BN=EM$, prove that $DF$ is perpendicular to $FG$.
2008 Harvard-MIT Mathematics Tournament, 4
Positive real numbers $ x$, $ y$ satisfy the equations $ x^2 \plus{} y^2 \equal{} 1$ and $ x^4 \plus{} y^4 \equal{} \frac {17}{18}$. Find $ xy$.
2017 CCA Math Bonanza, L5.3
How many ways are there to fill a $3\times3\times6$ rectangular prism with $1\times1\times2$ blocks? Rotations are not distinct. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\max\left(2\left(1-\left|\frac{C-A}{C}\right|\right),0\right)$.
[i]2017 CCA Math Bonanza Lightning Round #5.3[/i]
May Olympiad L2 - geometry, 2013.3
Many distinct points are marked in the plane. A student draws all the segments determined by those points, and then draws a line [i]r[/i] that does not pass through any of the marked points, but cuts exactly $60$ drawn segments. How many segments were not cut by [i]r[/i]? Give all possibilites.