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.

AND:
OR:
NO:

Found problems: 85335

1997 Israel National Olympiad, 3
Tags: product , number theory , inequalities , prime
Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

2007 Cono Sur Olympiad, 2
Tags: number theory
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

1972 IMO Longlists, 44
Tags: pigeonhole principle , combinatorics
Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

1950 Putnam, A5
Tags:
A function $D(n)$ of the positive integral variable $n$ is defined by the following properties: $D(1) = 0, D(p) = 1$ if $p$ is a prime, $D(uv) = u D(v) + v D(u)$ for any two positive integers $u$ and $v.$ Answer all three parts below. (i) Show that these properties are compatible and determine uniquely $D(n).$ (Derive a formula for $D(n) /n,$ assuming that $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ where $p_1, p_2, \ldots, p_k$ are different primes.) (ii) For what values of $n$ is $D(n) = n?$ (iii) Define $D^2 (n) = D[D(n)],$ etc., and find the limit of $D^m (63)$ as $m$ tends to $\infty.$

2019 IFYM, Sozopol, 7
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

2001 Federal Math Competition of S&M, Problem 1
Tags: derivative , calculus , number theory
Solve in positive integers \[ x^y + y = y^x + x \]

1993 All-Russian Olympiad, 3
Tags: quadratic , algebra
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

1985 AMC 12/AHSME, 19
Tags: quadratic , special factorizations , function
Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

2013 JBMO TST - Turkey, 5
Tags: inequalities
Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that \[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \] Prove that $x<y$.

VMEO II 2005, 8
Tags: inequalities
If a,b,c>0, prove that: \[ \frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}} \] thank u for ur help :oops:

KoMaL A Problems 2022/2023, A. 843
Tags: number theory
Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

2013 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , equation
Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

2006 USA Team Selection Test, 6
Tags: vector , circumcircle , geometry , trigonometry
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$

Durer Math Competition CD 1st Round - geometry, 2012.D2
Tags: geometry , area
Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half the area of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/e/7/a84ad28b3cd45bd0ce455cee2446222fd3eac2.png[/img]

2011 Poland - Second Round, 3
Tags: polynomial , integration , calculus , algebra
There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial: \[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\] is divisible by $P(x)-Q(x)$.

1995 Swedish Mathematical Competition, 5
Tags: cyclic quadrilateral , geometry , angle , cyclic
On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

PEN F Problems, 2
Tags: trigonometry , rational number , polynomial , algebra
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

2004 Gheorghe Vranceanu, 4
Tags: complex number , linear algebra , matrix
Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

2002 Junior Balkan Team Selection Tests - Romania, 4
Tags: area , geometric inequalities , combinatorial geometry , combinatorics , geometry
Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

1970 Spain Mathematical Olympiad, 4
Tags: polynomial , algebra
Knowing that the polynomials $$2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25$$ $$x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50$$ have two common double roots, determine all their roots.

2017 Argentina National Math Olympiad Level 2, 2
Tags: combinatorics
We say that a set of positive integers is [i]regular [/i] if, for any selection of numbers from the set, the sum of the chosen numbers is different from $1810$. Divide the set of integers from $452$ to $1809$ (inclusive) into the smallest possible number of regular sets.

2011 Romanian Master of Mathematics, 2
Tags: function , polynomial , system of equations , greatest common divisor , number theory , modular arithmetic , algebra
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

2015 BMT Spring, 6
Tags: polynomial , algebra
The roots of the equation $x^5-180x^4+Ax^3+Bx^2+Cx+D=0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

1965 Putnam, B5
Tags:
Consider collections of unordered pairs of $V$ different objects $a$, $b$, $c$, $\ldots$, $k$. Three pairs such as $ab$, $bc$, $ab$ are said to form a triangle. Prove that, if $4E\leq V^2$, it is possible to choose $E$ pairs so that no triangle is formed.

2010 Puerto Rico Team Selection Test, 3
Tags: combinatorics
Five children are divided into groups and in each group they take the hand forming a wheel to dance spinning. How many different wheels those children can form, if it is valid that there are groups of $1$ to $5$ children, and can there be any number of groups?