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

2008 Indonesia TST, 2
Tags: number theory , factorial , positive integer
Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

2019 AMC 12/AHSME, 21
Tags: complex number
Let $$z=\frac{1+i}{\sqrt{2}}.$$ What is $$(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?$$ $\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$

1990 AMC 12/AHSME, 1
Tags:
If $\dfrac{x/4}{2}=\dfrac{4}{x/2}$ then $x=$ $\textbf{(A) }\pm 1/2\qquad \textbf{(B) }\pm 1\qquad \textbf{(C) }\pm 2\qquad \textbf{(D) }\pm 4\qquad \textbf{(E) }\pm 8$

1991 Hungary-Israel Binational, 4
Tags: quadratic , system of equations , algebra
Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

Estonia Open Senior - geometry, 2018.1.1
Tags: geometry , lattice points , equilateral
Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

2001 Mexico National Olympiad, 3
Tags: geometry , cyclic quadrilateral , equal segments
$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.

2019 Durer Math Competition Finals, 1
Tags: algebra , sequence , sum , inequalities
Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

1967 IMO Shortlist, 6
Tags: inequality , polynomial , algebra , n-variable inequality
Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

2020 Turkey Junior National Olympiad, 1
Tags: quadratic equation , number theory
Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$

1985 IMO Longlists, 46
Tags: analytic geometry , geometry
Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$; t intersects $C$ at another point $Q$. Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$. Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$

2007 Kyiv Mathematical Festival, 4
Tags: geometric transformation , reflection , trapezoid , geometry
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

2020 China Northern MO, BP5
Tags: number theory , relatively prime , combinatorics , set
It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition $$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$ Determine the maximum positive integer $n$. *$(a,b)$ means $\gcd (a,b)$

2022 JHMT HS, 4
Tags: combinatorics
For a nonempty set $A$ of integers, let $\mathrm{range} \, A=\max A-\min A$. Find the number of subsets $S$ of \[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] such that $\mathrm{range} \, S$ is an element of $S$.

2022 Bangladesh Mathematical Olympiad, 8
Tags: number theory
Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

2006 Pre-Preparation Course Examination, 4
Tags: abstract algebra , algebra
If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?

2012 Portugal MO, 1
Tags: floor function , number theory
Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.

2020 Ukraine Team Selection Test, 2
Tags: algebra
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

2025 USA IMO Team Selection Test, 6
Tags: number theory
Prove that there exists a real number $\varepsilon>0$ such that there are infinitely many sequences of integers $0<a_1<a_2<\hdots<a_{2025}$ satisfying \[\gcd(a_1^2+1, a_2^2+1,\hdots, a_{2025}^2+1) > a_{2025}^{1+\varepsilon}.\] [i]Pitchayut Saengrungkongka[/i]

2020 USMCA, 6
Tags:
Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers?

2000 Harvard-MIT Mathematics Tournament, 27
Tags:
What is the smallest number that can be written as a sum of $2$ squares in $3$ ways?

1996 Tournament Of Towns, (517) 4
Tags: combinatorics
For what integers $n > 1$ can it happen that in a group of $n +1$ girls and $n$ boys, all the girls know a different number of boys while all the boys know the same number of girls? (NB Vassiliev)

2021 Alibaba Global Math Competition, 15
Tags: calculus , integration , topology , analysis , manifold , differential geometry
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

PEN J Problems, 13
Tags: divisor function
Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.

2019 Junior Balkan MO, 1
Tags: number theory , prime number
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Novosibirsk Oral Geo Oly VIII, 2020.5
Tags: perpendicular bisector , midpoint , geometry
Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.