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

1995 AMC 12/AHSME, 13
Tags:
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$. \[\begin{tabular}{ccccccc}& 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0\\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}\] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{More than } 10$

2009 Romania National Olympiad, 2
Tags: algebra
Show that for any four positive real numbers $ a,b,c,d $ and four negative real numbers $ e,f,g,h, $ the terms $ ae+bc,ef+cg,fd+gh,da+hb $ are not all positive.

2009 Today's Calculation Of Integral, 488
Tags: integration , logarithm , calculus , calculus computations , inequalities , trigonometry
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$

2006 Bosnia and Herzegovina Team Selection Test, 3
Tags: positive integer , fractional part , inequality , inequalities , number theory
Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

PEN H Problems, 77
Tags: diophantine equation , relatively prime , number theory , ratio
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2019 Switzerland - Final Round, 7
Tags: angle , equal segments , equal angles , geometry
Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

2017 AMC 12/AHSME, 10
Tags: probability
Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number? $\textbf{(A)}~\frac12 \qquad \textbf{(B)}~\frac23 \qquad \textbf{(C)}~\frac34 \qquad \textbf{(D)}~\frac56\qquad \textbf{(E)}~\frac78$

1960 AMC 12/AHSME, 23
Tags: integration , calculus
The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by: $ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad$ $\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad$ $\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad$ $\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad$ $\textbf{(E)}\ \text{two real values of} \text{ } x $

2008 ISI B.Stat Entrance Exam, 7
Tags: algebra , function
Consider the equation $x^5+x=10$. Show that (a) the equation has only one real root; (b) this root lies between $1$ and $2$; (c) this root must be irrational.

KoMaL A Problems 2019/2020, A. 778
Tags: diophantine equation , number theory
Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying $x^2+dy^2=2^n$ Submitted by Kada Williams, Cambridge

2008 Princeton University Math Competition, A2/B3
Tags: combinatorics
Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infi nity, what is the ratio between the number of triangles and the number of squares?

1963 AMC 12/AHSME, 11
Tags:
The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is: $\textbf{(A)}\ 38.5 \qquad \textbf{(B)}\ 37.5 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 36.5 \qquad \textbf{(E)}\ 36$

1996 Iran MO (3rd Round), 3
Tags: geometry
Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.

2006 Junior Tuymaada Olympiad, 6
Tags: sum , combinatorics , number theory
[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

1996 Baltic Way, 15
Tags: inequalities
For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?

MathLinks Contest 4th, 7.1
Tags: inequalities , algebra
Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that $$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$

STEMS 2024 Math Cat A, P2
Tags: combinatorics
Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

2020 HMIC, 4
Tags: catalan , polynomial , number theory , algebra
Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set \[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\] are divisible by $p$. [i]Tristan Shin[/i]

2020 Polish Junior MO First Round, 2.
Tags: easy , geometry
Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

Russian TST 2018, P3
Tags: inequalities
Let $a,b,c>0.$ Prove that $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \ge \frac{1}{\sqrt{2a^2+2bc}}+\frac{1}{\sqrt{2b^2+2ca}}+\frac{1}{\sqrt{2c^2+2ab}}$

2017 USA TSTST, 6
Tags: number theory , fibonacci , algebra
A sequence of positive integers $(a_n)_{n \ge 1}$ is of [i]Fibonacci type[/i] if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences? [i]Proposed by Ivan Borsenco[/i]

2024 AMC 8 -, 8
Tags:
On Monday Taye has \$2. Everyday he either gains \$3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later? $\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

1972 Putnam, A5
Tags: divisible , number theory
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

2015 ASDAN Math Tournament, 3
Tags: geometry test
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.

1986 IMO Longlists, 49
Tags: radical axis , rectangle , geometry , power of a point , induction
Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows: $C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.