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

2010 Today's Calculation Of Integral, 534
Tags: calculus , function , integration , calculus computations
Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.

1973 AMC 12/AHSME, 5
Tags:
Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), I. Averaging is associative II. Averaging is commutative III. Averaging distributes over addition IV. Addition distributes over averaging V. Averaging has an identity element those which are always true are $ \textbf{(A)}\ \text{All} \qquad \textbf{(B)}\ \text{I and II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad \textbf{(D)}\ \text{II and IV only} \qquad \textbf{(E)}\ \text{II and V only}$

2003 AMC 12-AHSME, 16
Tags: analytic geometry , symmetry , geometry , probability
A point $ P$ is chosen at random in the interior of equilateral triangle $ ABC$. What is the probability that $ \triangle ABP$ has a greater area than each of $ \triangle ACP$ and $ \triangle BCP$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{2}{3}$

1951 Miklós Schweitzer, 15
Tags: geometric transformation , advanced fields , rotation , geometry
Let the line $ z\equal{}x, \, y\equal{}0$ rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.

2008 Hong Kong TST, 2
Tags: limit , inequalities
Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

2006 Baltic Way, 1
Tags:
For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$. What is the largest number of its consecutive elements that can all be positive?

2000 Federal Competition For Advanced Students, Part 2, 3
Tags: function , algebra
Find all functions $f : \mathbb R \to \mathbb R$ such that for all reals $x, y, z$ it holds that \[f(x + f(y + z)) + f(f(x + y) + z) = 2y.\]

2001 Federal Math Competition of S&M, Problem 1
Tags: number theory
Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.

2000 Tuymaada Olympiad, 5
Tags: number theory , prime , divisible
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

2011 Today's Calculation Of Integral, 736
Tags: derivative , calculus computations , calculus , logarithm , integration
Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

1999 Canada National Olympiad, 2
Tags: geometry
Let $ABC$ be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$. Prove that the arc of the circle that is inside the triangle always has the same length.

2018 Canadian Mathematical Olympiad Qualification, 1
Tags: algebra , system of equations
Determine all real solutions to the following system of equations: $$ \begin{cases} y = 4x^3 + 12x^2 + 12x + 3\\ x = 4y^3 + 12y^2 + 12y + 3. \end{cases} $$

2005 iTest, 39
Tags: sum of powers , number theory
What is the smallest positive integer that when raised to the $6^{th}$ power, it can be represented by a sum of the $6^{th}$ powers of distinct smaller positive integers?

2022 USA TSTST, 2
Tags: geometry
Let $ABC$ be a triangle. Let $\theta$ be a fixed angle for which \[\theta<\frac12\min(\angle A,\angle B,\angle C).\] Points $S_A$ and $T_A$ lie on segment $BC$ such that $\angle BAS_A=\angle T_AAC=\theta$. Let $P_A$ and $Q_A$ be the feet from $B$ and $C$ to $\overline{AS_A}$ and $\overline{AT_A}$ respectively. Then $\ell_A$ is defined as the perpendicular bisector of $\overline{P_AQ_A}$. Define $\ell_B$ and $\ell_C$ analogously by repeating this construction two more times (using the same value of $\theta$). Prove that $\ell_A$, $\ell_B$, and $\ell_C$ are concurrent or all parallel.

2012 Bogdan Stan, 1
Tags: inequalities , algebra
Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that $$ a^2+b^2+c^2\le 2, $$ and determine the cases in which equality is attained.

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: prime , consecutive , divide , sum of powers , number theory
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

2020 HMNT (HMMO), 1
Tags: floor function , algebra
For how many positive integers $n \le 1000$ does the equation in real numbers $x^{\lfloor x \rfloor } = n$ have a positive solution for $x$?

2007 Tournament Of Towns, 5
Tags:
Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible). [i](4 points)[/i]

2022 AMC 12/AHSME, 2
Tags: algebra
The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers? $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$

Kyiv City MO 1984-93 - geometry, 1991.9.3
Tags: geometry , area
The point $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of ​​which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of ​​the triangle $BKM$.

2006 Harvard-MIT Mathematics Tournament, 4
Tags:
A dot is marked at each vertex of a triangle $ABC$. Then, $2$, $3$, and $7$ more dots are marked on the sides $AB$, $BC$, and $CA$, respectively. How many triangles have their vertices at these dots?

Kvant 2025, M2834
Tags: algebra
Let's call a set of numbers [i]lucky[/i] if it cannot be divided into two nonempty groups so that the product of the sum of the numbers in one group and the sum of the numbers in the other is positive. The teacher wrote several integers on the blackboard. Prove that the children can add another integer to the existing ones so that the resulting set is lucky. [i]A. Kuznetsov[/i]

1997 Vietnam Team Selection Test, 2
Tags: logarithm , algebra
Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.

II Soros Olympiad 1995 - 96 (Russia), 11.4
Tags: analysis , calculus , algebra
Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.

2004 Germany Team Selection Test, 2
Tags: angle bisector , geometry , menelaus
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.