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

1999 Tournament Of Towns, 7
Tags: convex polyhedron , polyhedron , combinatorics , combinatorial geometry
Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides. (A Kanel)

2010 Princeton University Math Competition, 2
Tags:
Consider the following two-player game: player $A$ (first mover) and $B$ take turns to write a positive integer less than or equal to $10$ on the blackboard. The integer written at any step cannot be a factor of any existing integer on board. Determine, with proof, who wins.

2018 Polish Junior MO First Round, 2
Tags: geometry , parallelogram , Poland
Inside parallelogram $ABCD$ is point $P$, such that $PC = BC$. Show that line $BP$ is perpendicular to line which connects middles of sides of line segments $AP$ and $CD$.

2007 Putnam, 2
Tags: Putnam , geometry , conics , hyperbola , analytic geometry , inequalities , rectangle
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)

2001 JBMO ShortLists, 5
Tags: number theory proposed , number theory
Let $x_k=\frac{k(k+1)}{2}$ for all integers $k\ge 1$. Prove that for any integer $n \ge 10$, between the numbers $A=x_1+x_2 + \ldots + x_{n-1}$ and $B=A+x_n$ there is at least one square.

PEN H Problems, 58
Tags: modular arithmetic , algorithm , Diophantine Equations
Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.

2006 Turkey Junior National Olympiad, 2
Tags: calculus , integration
Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

2016 JBMO Shortlist, 2
Tags: JBMO , combinatorics , Sum , prime
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

2019 Romania National Olympiad, 4
Tags:
Let $A$ and $B$ be two nonempty finite sets of nonnegative integers. We denote by $\mathcal{F}$ the set of all functions $f:\mathcal{P}(A) \to B$ that satisfy [center]$f(X\cap Y)=\min \{f(X), f(Y)\},$ for all $X,Y \subset A,$[/center] and by $\mathcal{G}$ the set of all functions $g:\mathcal{P}(A) \to B$ that satisfy [center]$g(X\cup Y)=\max \{g(X), g(Y)\},$ for all $X,Y \subset A.$[/center] Prove that $\mathcal F$ and $\mathcal G$ have the same number of elements and find this number.

1998 Czech And Slovak Olympiad IIIA, 2
Tags: Sum , algebra , Subsets
Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$

1970 Polish MO Finals, 6
Tags: quadratics , algebra , inequalities
Find the smallest real number $A$ such that, for every quadratic polynomial $f(x)$ satisfying $ | f(x)| \le 1$ for $0 \le x \le 1$, it holds that $f' (0) \le A$.

1991 Balkan MO, 2
Tags: geometry , number theory , relatively prime , geometry open
Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.

2009 All-Russian Olympiad Regional Round, 11.4
Tags: geometry , midline
In an acute non-isosceles triangle $ABC$, the altitude $AA'$ is drawn and point $H$ is the intersection point of the altitudes and and $O$ is the center of the circumscribed circle. Prove that the point symmetric to the circumcenter of triangle $HOA'$ wrt straight line $HO$, lies on a midline of triangle $ABC$.

2008 South africa National Olympiad, 5
Tags: geometry , circumcircle , rectangle , geometry unsolved
Triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)

1993 IMO Shortlist, 7
Tags: polynomial , algebra , Irreducible , factoring polynomials , IMO , IMO 1993 , irreducible polynomial
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

1990 Greece National Olympiad, 3
Tags: function , algebra , inequalities , functional
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satisfy $y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) $ for any $x,y \in\mathbb{R}$.

JOM 2015 Shortlist, C3
Tags: function , combinatorics
Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy: i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\ ii) $ f(g(x))=f(x) , \forall x \in S $\\ iii) $f(x+n)=f(x) ,\forall x \in S$\\ iv) $ g $ is bijective.\\ Find the number of pair of such functions $ (f,g)$ for every $n$.

1993 Denmark MO - Mohr Contest, 3
Tags: algebra , system of equations , System
Determine all real solutions $x,y$ to the system of equations $$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$

2002 AMC 12/AHSME, 11
Tags:
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$

2016 Purple Comet Problems, 27
Tags: Purple Comet
A container the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the container. Find the volume of the liquid in the container. [center][img]https://snag.gy/CdvpUq.jpg[/img][/center]

2021 Israel National Olympiad, P1
Tags: combinatorics , number theory , Digits
Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

2010 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.

2000 Brazil Team Selection Test, Problem 2
Tags: number theory
For a positive integer $n$, let $A_n$ be the set of all positive numbers greater than $1$ and less than $n$ which are coprime to $n$. Find all $n$ such that all the elements of $A_n$ are prime numbers.

2014 Czech-Polish-Slovak Junior Match, 6
Tags: inequalities , algebra , minimum , maximum , min , max
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.

2023 Indonesia Regional, 4
Tags: irrational number , algebra , Indonesia , RMO
Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \] are both rational numbers.