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

1997 Tuymaada Olympiad, 6
Tags: divisor , consecutive , number theory
Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

Fractal Edition 1, P3
Tags: inequalities
Let \( a \), \( b \), and \( c \) be three positive real numbers that satisfy \( ab + bc + ca = 1 \). Show that: \[ \frac{a}{a^2+1} + \frac{b}{b^2+1} + \frac{c}{c^2+1} \le \frac{1}{4abc} \]

2024 HMNT, 10
Tags:
Let $S = \{1, 2, 3, . . . , 64\}.$ Compute the number of ways to partition $S$ into $16$ arithmetic sequences such that each arithmetic sequence has length $4$ and common difference $1, 4,$ or $16.$

2016 ASDAN Math Tournament, 2
Tags:
Define a $\textit{subsequence}$ of a string $\mathcal{S}$ of letters to be a positive-lenght string using any number of the letters in $\mathcal{S}$ in order. For example, a subsequence of $HARRISON$ is $ARRON$. Compute the number of subsequences in $HARRISON$.

2015 CCA Math Bonanza, L5.4
Tags:
Submit a positive integer $x$ between $1$ and $10$ inclusive. Your score on the problem will be proportional to \[ \frac{11-x}{n} \] where $n$ is the number of teams that also submit the number $x$. [i]2015 CCA Math Bonanza Lightning Round #5.4[/i]

2006 Stanford Mathematics Tournament, 25
Tags:
For positive integers $ n$ let $ D(n)$ denote the set of positive integers that divide $ n$ and let $ S(n)\equal{}\Sigma_{k \in D(n)} \frac{1}{k}$. What is $ S(2006)$? Answer with a fraction reduced to lowest terms.

2007 ITest, 4
Tags: probability
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice. $\textbf{(A) }\dfrac18\hspace{14em}\textbf{(B) }\dfrac3{16}\hspace{14em}\textbf{(C) }\dfrac38$ $\textbf{(D) }\dfrac12$

2013 AMC 8, 11
Tags:
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2000 Romania National Olympiad, 3
Tags: 3d geometry , pyramid , geometry
Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation $$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$

2005 Argentina National Olympiad, 3
Tags: floor function , algebra
Let $a$ be a real number such that $\frac{1}{a}=a-[a]$. Show that $a$ is irrational. Clarification: The brackets indicate the integer part of the number they enclose.

2002 Manhattan Mathematical Olympiad, 1
Tags: modular arithmetic
Prove that if an integer $n$ is of the form $4m+3$, where $m$ is another integer, then $n$ is not a sum of two perfect squares (a perfect square is an integer which is the square of some integer).

2020 AMC 10, 11
Tags: median
What is the median of the following list of $4040$ numbers$?$ $$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$ $\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$

2021 USMCA, 12
Tags:
Find the sum of the three smallest positive integers $N$ such that $N$ has a units digit of $1,$ $N^2$ has a tens digit of $2,$ and $N^3$ has a hundreds digit of $3.$

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: equal angles , tangent , geometry , circles
Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

2016 ASDAN Math Tournament, 10
Tags:
Let $\mathcal{S}$ be the set of all possible $9$-digit numbers that use $1,2,3,\dots,9$ each exactly once as a digit. What is the probability that a randomly selected number $n$ from $\mathcal{S}$ is divisible by $27$?

2019 Online Math Open Problems, 8
Tags:
In triangle $ABC$, side $AB$ has length $10$, and the $A$- and $B$-medians have length $9$ and $12$, respectively. Compute the area of the triangle. [i]Proposed by Yannick Yao[/i]

2012 Romania Team Selection Test, 1
Tags: quadratic , number theory , algebra , polynomial
Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

2024/2025 TOURNAMENT OF TOWNS, P1
Tags: number theory
Find the minimum positive integer such that some four of its natural divisors sum up to $2025$.

1972 IMO Longlists, 45
Tags: geometry
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.

2006 AMC 10, 11
Tags: factorial
What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$? $ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$

2018 Latvia Baltic Way TST, P2
Tags: system of equations , algebra
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations: $$\begin{cases} y(x+y)^2=2\\ 8y(x^3-y^3) = 13. \end{cases}$$

2017 Moldova Team Selection Test, 9
Tags: algebra
Let $$P(X)=a_{0}X^{n}+a_{1}X^{n-1}+\cdots+a_{n}$$ be a polynomial with real coefficients such that $a_{0}>0$ and $$a_{n}\geq a_{i}\geq 0,$$ for all $i=0,1,2,\ldots,n-1.$ Prove that if $$P^{2}(X)=b_{0}X^{2n}+b_{1}X^{2n-1}+\cdots+b_{n-1}X^{n+1}+\cdots+b_{2n},$$ then $P^2(1)\geq 2b_{n-1}.$

2024 JHMT HS, 3
Tags: number theory
Let $N_2$ be the answer to problem 2. On a number line, Tanya circles the first $\ell$ positive integers. Then, starting with the greatest number in the most recent circle, she circles the next $\ell$ positive integers, so that the two circles have exactly one number in common; she repeats this until $N_2$ is in a circle. Compute the sum of all possible values of $\ell$ for which $N_2$ is the greatest number in a circle.

2002 China Team Selection Test, 2
Tags: circumcircle , incenter , reflection , geometry , parallelogram , inequalities , geometric transformation
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

1979 Miklós Schweitzer, 5
Tags: advanced fields
Give an example of ten different noncoplanar points $ P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $ 3$-space such that connecting each $ P_i$ to each $ Q_j$ by a rigid rod results in a rigid system. [i]L. Lovasz[/i]