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

OMMC POTM, 2024 1

Luke chose a set of three different dates $a,b,c$ in the month of May, where in any year, if one makes a calendar with a sheet of grid paper the centers of the cells with dates $a,b,c$ would form an isosceles right triangle or a straight line. How many sets can be chosen? [img]https://cdn.artofproblemsolving.com/attachments/7/3/dbf90fdc81fc0f0d14c32020b69df53b67b397.png[/img]

2017 AIME Problems, 14

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Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.

2004 Flanders Junior Olympiad, 1

Two $5\times1$ rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]

2024 Mexico National Olympiad, 2

Determine all pairs $(a, b)$ of integers that satisfy both: 1. $5 \leq b < a$ 2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order. [b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.

1979 IMO Longlists, 35

Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

2007 ISI B.Stat Entrance Exam, 2

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

2008 Bulgaria Team Selection Test, 2

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

2015 HMMT Geometry, 1

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Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

2004 Chile National Olympiad, 6

The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.

Putnam 1939, A4

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Given $4$ lines in Euclidean $3-$space: $L_1: x = 1, y = 0;$ $L_2: y = 1, z = 0;$ $L_3: x = 0, z = 1;$ $L_4: x = y, y = -6z.$ Find the equations of the two lines which both meet all of the $L_i.$

1996 Korea National Olympiad, 2

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

1999 Baltic Way, 4

For all positive real numbers $x$ and $y$ let \[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \] Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.

2023 CMIMC Geometry, 10

Tags: geometry
The vertices of $\triangle ABC$ are labeled in counter-clockwise order, and its sides have lengths $CA = 2022$, $AB = 2023$, and $BC = 2024$. Rotate $B$ $90^\circ$ counter-clockwise about $A$ to get a point $B'$. Let $D$ be the orthogonal projection of $B'$ unto line $AC$, and let $M$ be the midpoint of line segment $BB'$. Then ray $BM$ intersects the circumcircle of $\triangle CDM$ at a point $N \neq M$. Compute $MN$. [i]Proposed by Thomas Lam[/i]

2004 China Team Selection Test, 1

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

2004 China Team Selection Test, 1

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

1972 IMO Longlists, 33

A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$. $(a)$ Prove that \[U(n + 1)+U(n -1) = 4U(n);\] $(b)$ Prove that \[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]

2020-2021 OMMC, 4

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Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

2004 IMC, 4

Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

2022 Miklós Schweitzer, 3

Original in Hungarian; translated with Google translate; polished by myself. Let $f: [0, \infty) \to [0, \infty)$ be a function that is linear between adjacent integers, and for $n = 0, 1, \dots$ satisfies $$f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}$$ Let $f^1(x) = f(x)$, and $f^k(x) = f(f^{k - 1}(x))$ for all integers $k \geq 2$. Determine the values of $\liminf\nolimits_{k\to\infty}f^k(x)$ and $\limsup\nolimits_{k\to\infty}f^k(x)$ for almost all $x \in [0, \infty)$ under Lebesgue measure. (Not sure whether the last sentence translates correctly; the original: Határozzuk meg Lebesgue majdnem minden $x\in [0, \infty)$-re a $\liminf\nolimits_{k\to\infty}f^k(x)$ és $\limsup\nolimits_{k\to\infty}f^k(x)$ értékét.)

2002 Iran Team Selection Test, 2

$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

1974 IMO Longlists, 29

Tags: geometry
Let $A,B,C,D$ be points in space. If for every point $M$ on the segment $AB$ the sum \[S_{AMC}+S_{CMD}+S_{DMB}\] Is constant show that the points $A,B,C,D$ lie in the same plane. [hide="Note."] [i]Note. $S_X$ denotes the area of triangle $X.$[/i][/hide]

2024 Sharygin Geometry Olympiad, 10.8

The common tangents to the circumcircle and an excircle of triangle $ABC$ meet $BC, CA,AB$ at points $A_1, B_1, C_1$ and $A_2, B_2, C_2$ respectively. The triangle $\Delta_1$ is formed by the lines $AA_1, BB_1$, and $CC_1$, the triangle $\Delta_2$ is formed by the lines $AA_2, BB_2,$ and $CC_2$. Prove that the circumradii of these triangles are equal.

2012 Grigore Moisil Intercounty, 2

[b]a)[/b] Prove that $$ k+\frac{1}{2}-\frac{1}{8k}<\sqrt{k^2+k}<k+\frac{1}{2}-\frac{1}{8k}+\frac{1}{16k^2} , $$ for any natural number $ k. $ [b]b)[/b] Prove that there exists four numbers $ \alpha,\beta,\gamma,\delta\in\{0,1,2,3,4,5,6,7,8,9\} $ such that $$ \left\lfloor\sum_{k=1}^{2012} \sqrt{k(k+1)\left( k^2+k+1 \right)}\right\rfloor =\underbrace{\ldots\alpha \beta\gamma\delta}_{\text{decimal form}} $$ and $ \alpha +\delta =\gamma . $

2017 NIMO Summer Contest, 6

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Let $P = (-2, 0)$. Points $P$, $Q$, $R$ lie on the graph of the function $y = x^3 - 3x + 2$ such that $Q$ is the midpoint of segment $PR$. Compute $PR^2$. [i]Proposed by David Altizio[/i]

2006 Romania National Olympiad, 1

Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$. [i]Dan Schwarz[/i]