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

2000 Moldova National Olympiad, Problem 6

A natural number $n\ge5$ leaves the remainder $2$ when divided by $3$. Prove that the square of $n$ is not a sum of a prime number and a perfect square.

2002 APMO, 1

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

2014 Putnam, 6

Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$

2003 China Western Mathematical Olympiad, 1

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

1974 IMO Shortlist, 8

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

1969 Czech and Slovak Olympiad III A, 2

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

1993 AMC 8, 1

Tags:
Which pair of numbers does NOT have a product equal to $36$? $\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$

2022 MIG, 14

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Four coins are placed in a line. A passerby walks by and flips each coin, and stops if she ever obtains two adjacent heads. If the passerby manages to flip all four coins, how many possible head-tail combinations exist for her four flips? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }12$

2006 China Second Round Olympiad, 3

Solve the system of equations in real numbers: \[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]

2024 Thailand TST, 2

Tags: nithi , mmp , incenter , geometry
Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$. Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.

2005 AMC 10, 4

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For real numbers $ a$ and $ b$, define $ a \diamond b \equal{} \sqrt{a^2 \plus{} b^2}$. What is the value of \[(5\diamond 12)\diamond ((\minus{}12) \diamond (\minus{}5))?\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{17}{2}\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 13\sqrt{2}\qquad \textbf{(E)}\ 26$

2015 BMT Spring, 13

There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?

2016 Tournament Of Towns, 1

All integers from $1$ to one million are written on a tape in some arbitrary order. Then the tape is cut into pieces containing two consecutive digits each. Prove that these pieces contain all two-digit integers for sure, regardless of the initial order of integers.[i](4 points)[/i] [i]Alexey Tolpygo[/i]

1957 Moscow Mathematical Olympiad, 348

A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

1998 Iran MO (3rd Round), 2

Let $ABCD$ be a convex pentagon such that \[\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.\] Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that \[\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.\]

1972 Czech and Slovak Olympiad III A, 5

Determine how many unordered pairs $\{A,B\}$ is there such that $A,B\subseteq\{1,\ldots,n\}$ and $A\cap B=\emptyset.$

1988 IMO Shortlist, 28

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

2024 CCA Math Bonanza, L5.4

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Answer this question with a positive integer $1$ through $1000$. A positive integer ``answer" has been randomly selected from 1 to 1000, inclusive; if your selected integer is less than or equal to the ``answer", you will gain $\lfloor 20\left(\frac{x}{a}\right)^2 \rfloor$ points, where $x$ is your number and $a$ is the ``answer". If you select an integer greater than the ``answer", you will not gain any points. [i]Lightning 5.4[/i]

2013 Spain Mathematical Olympiad, 4

Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.

2021 CMIMC, 1.6

Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$. [i]Proposed by Vijay Srinivasan[/i]

1990 Chile National Olympiad, 5

Tags: sum , algebra
Determine a natural $n$ such that $$996 \le \sum_{k = 1}^{n}\frac{1}{k}$$

2002 Dutch Mathematical Olympiad, 1

The sides of a $10$ by $10$ square $ABCD$ are reflective on the inside. A beam of light enters the square via the vertex $A$ and heads to the point $P$ on $CD$ with $CP = 3$ and $PD = 7$. In $P$ it naturally reflects on the $CD$ side. The light beam can only leave the square via one of the angular points $A, B, C$ or $D$. What is the distance that the light beam travels within the square before it leaves the square again? By which vertex does that happen?

LMT Theme Rounds, 9

Tags:
A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Nathan Ramesh

2018 Bulgaria National Olympiad, 3.

Prove that \[ \left(\frac{6}{5}\right)^{\sqrt{3}}>\left(\frac{5}{4}\right)^{\sqrt{2}}. \]

2018 Turkey Team Selection Test, 1

Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.