This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2021 Azerbaijan Junior NMO, 3
Tags: inequality
$a,b,c $ are positive real numbers . Prove that $\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$

2022 Rioplatense Mathematical Olympiad, 5
Tags: geometry
Let $ABCDEFGHI$ be a regular polygon with $9$ sides and the vertices are written in the counterclockwise and let $ABJKLM$ be a regular polygon with $6$ sides and the vertices are written in the clockwise. Prove that $\angle HMG=\angle KEL$. Note: The polygon $ABJKLM$ is inside of $ABCDEFGHI$.

2024 Canada National Olympiad, 1
Tags: geometry , incenter , geometric transformation , reflection , angle bisector
Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

2017 China Team Selection Test, 1
Tags: combinatorics
Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

2016 Purple Comet Problems, 2
Tags:
The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles. [center] [img]https://i.snag.gy/BlqcSQ.jpg[/img] [/center]

2015 Peru IMO TST, 11
Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

2024 Turkey Olympic Revenge, 1
Tags: combinatorics
Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal. Proposed by [i]Sena Başaran[/i]

2017 Spain Mathematical Olympiad, 5
Tags: algebra , inequalities , maximum
Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

1996 Bulgaria National Olympiad, 3
Tags: combinatorics
A square table of size $7\times 7$ with the four corner squares deleted is given. [list=a] [*] What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table? [*] Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [/list] [asy] size(3.5cm); usepackage("amsmath"); MP("\text{Figure }1.", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy]

PEN E Problems, 18
Tags:
Without using Dirichlet's theorem, show that there are infinitely many primes ending in the digit $9$.

2007 Regional Olympiad of Mexico Center Zone, 4
Tags: number theory , power of 2
Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

1968 Putnam, A5
Tags: polynomial
Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$.

2023 China Girls Math Olympiad, 6
Tags: algebra
Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of $$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$

2003 Iran MO (2nd round), 1
Tags: number theory
We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

2010 Contests, 3
Tags:
In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.

2019 Iran MO (3rd Round), 3
Tags: rectangle , combinatorics
Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.

2017 Brazil Team Selection Test, 2
Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2023 Israel National Olympiad, P1
Tags: combinatorics
2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

2008 JBMO Shortlist, 4
Tags: algebra , system of equations
Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

2018 AMC 8, 5
Tags:
What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$? $\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$

2011 Germany Team Selection Test, 2
Tags: geometry , circumcircle , reflection , parallelogram
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

1989 Tournament Of Towns, (235) 3
Tags: number theory , perfect square , combinatorics
Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

1996 All-Russian Olympiad Regional Round, 8.1
Tags: algebra
Ice cream costs $2000$ rubles. Petya has $$400^5 - 399^2\cdot (400^3 + 2\cdot 400^2 + 3\cdot 400 + 4)$$ rubles. Does Petya have enough money for ice cream?

1993 AIME Problems, 6
Tags: number theory , least common multiple
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

2012-2013 SDML (Middle School), 6
Tags:
Triangle $ABC$ has right angle $A$, with $AB=AC=6$. Point $P$ is located within triangle $ABC$ such that the areas of triangles $ABP$, $BCP$, and $ACP$ are equal. Express $CP$ in simplest radical form.