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

1971 IMO Longlists, 30
Tags: parameterization , system of equations , algebra
Prove that the system of equations \[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \] $a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?

2008 ITest, 84
Tags: algebra , polynomial
Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.

2020 BMT Fall, 8
Tags: algebra , inequalities
Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$.

2012 Tournament of Towns, 5
Tags: reflection , angle bisector , congruent triangles , incircle , tangent , geometry
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.

2024 HMNT, 13
Tags: guts
Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$

2018 AMC 10, 4
Tags:
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

2010 Harvard-MIT Mathematics Tournament, 3
Tags:
Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\ldots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}<i$ and $-1$ if $S_{i-1}\geq i$. What is the largest $k\leq 2010$ such that $S_k=0$?

2018 Stanford Mathematics Tournament, 7
Tags: geometry , 3d geometry
Two equilateral triangles $ABC$ and $DEF$, each with side length $1$, are drawn in $2$ parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon $AF BDCE$. Line segments $AE$, $AF$, $BF$, $BD$, $CD,$ and $CE$ are drawn, and suppose that each of these segments also has length $1$. Compute the volume of the resulting solid that is formed.

2023 Romania National Olympiad, 1
Tags: number theory , divisibility
The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.

2005 Junior Balkan Team Selection Tests - Romania, 18
Tags: algebra
Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

2023 VN Math Olympiad For High School Students, Problem 4
Tags: algebra , polynomial
Prove that: a polynomial is irreducible in $\mathbb{Z}[x]$ if and only if it is irreducible in $\mathbb{Q}[x].$

2008 Mexico National Olympiad, 2
Tags: geometry , 3d geometry , greatest common divisor , number theory
We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices. a) Prove that $\frac23E\le V$. b) Can $E=V$?

2021-IMOC qualification, A3
Tags: function , functional inequality , functional , inequalities , algebra
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$

2006 Harvard-MIT Mathematics Tournament, 1
Tags: geometry , perimeter
Octagon $ABCDEFGH$ is equiangular. Given that $AB=1$, $BC=2$, $CD=3$, $DE=4$, and $EF=FG=2$, compute the perimeter of the octagon.

2018 HMNT, 1
Tags: geometry
Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?

2002 IMC, 7
Tags: linear algebra , matrix
Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$, $$a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if}\, i\ne j,\\ 2 & \text{if}\, i= j. \end{cases}$$

2006 China Second Round Olympiad, 1
Tags:
Let $\triangle ABC$ be a given triangle. If $|BA-tBC| \ge |AC|$ for any $t \in \mathbb{R}$, then $\triangle ABC$ is $ \textbf{(A)}\ \text{an acute triangle}\qquad\textbf{(B)}\ \text{an obtuse triangle}\qquad\textbf{(C)}\ \text{a right triangle}\qquad\textbf{(D)}\ \text{not known}\qquad$

1964 All Russian Mathematical Olympiad, 050
Tags: geometry , tangential
The quadrangle $ABCD$ is circumscribed around the circle with the centre $O$. Prove that $$\angle AOB+ \angle COD=180^o. $$

2003 All-Russian Olympiad Regional Round, 10.3
Tags: combinatorics
$45$ people came to the alumni meeting. It turned out that any two of them, having the same number of acquaintances among those who came, don't know each other. What is the greatest number of pairs of acquaintances that could to be among those participating in the meeting?

1960 Miklós Schweitzer, 3
Tags:
[b]3.[/b] Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let $\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$. Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .[b](f. 6)[/b]

2009 All-Russian Olympiad, 8
Tags: number theory
Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.

2023 Kyiv City MO Round 1, Problem 4
Tags: combinatorics
For $n \ge 2$ consider $n \times n$ board and mark all $n^2$ centres of all unit squares. What is the maximal possible number of marked points that we can take such that there don't exist three taken points which form right triangle? [i]Proposed by Mykhailo Shtandenko[/i]

1981 IMO, 2
Tags: geometry , incenter , circumcircle , similar triangles
Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

2023 LMT Fall, 21
Tags: algebra , number theory , combinatorics
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$ [i]Proposed by Muztaba Syed[/i]

2021 CCA Math Bonanza, T2
Tags:
Given that real numbers $a$, $b$, and $c$ satisfy $ab=3$, $ac=4$, and $b+c=5$, the value of $bc$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]2021 CCA Math Bonanza Team Round #2[/i]