This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 85335

2022 IMO Shortlist, A1

Tags: sequence , algebra
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

1998 Estonia National Olympiad, 2

Let $C$ and $D$ be two distinct points on a semicircle of diameter $AB$. Let $E$ be the intersection of $AC$ and $BD$, $F$ be the intersection of $AD$ and $BC$ and $X, Y$, and $Z$ are the midpoints of $AB, CD$, and $EF$, respectively. Prove that the points $X, Y,$ and $Z$ are collinear.

2020 Estonia Team Selection Test, 3

The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

2006 District Olympiad, 2

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

2013 Putnam, 1

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

EMCC Guts Rounds, 2018

[u]Round 5[/u] [b]p13.[/b] Find all ordered pairs of real numbers $(x, y)$ satisfying the following equations: $$\begin{cases} \dfrac{1}{xy} + \dfrac{y}{x}= 2 \\ \dfrac{1}{xy^2} + \dfrac{y^2}{x} = 7 \end{cases}$$ [b]p14.[/b] An egg plant is a hollow prism of negligible thickness, with height $2$ and an equilateral triangle base. Inside the egg plant, there is enough space for four spherical eggs of radius $1$. What is the minimum possible volume of the egg plant? [b]p15.[/b] How many ways are there for Farmer James to color each square of a $2\times 6$ grid with one of the three colors eggshell, cream, and cornsilk, so that no two adjacent squares are the same color? [u]Round 6[/u] [b]p16.[/b] In a triangle $ABC$, $\angle A = 45^o$, and let $D$ be the foot of the perpendicular from $A$ to segment $BC$. $BD = 2$ and $DC = 4$. Let $E$ be the intersection of the line $AD$ and the perpendicular line from $B$ to line $AC$. Find the length of $AE$. [b]p17.[/b] Find the largest positive integer $n$ such that there exists a unique positive integer $m$ satisfying $$\frac{1}{10} \le \frac{m}{n} \le \frac19$$ [b]p18.[/b] How many ordered pairs $(A,B)$ of positive integers are there such that $A+B = 10000$ and the number $A^2 + AB + B$ has all distinct digits in base $10$? [u]Round 7[/u] [b]p19.[/b] Pentagon $JAMES$ satisfies $JA = AM = ME = ES = 2$. Find the maximum possible area of $JAMES$. [b]p20.[/b] $P(x)$ is a monic polynomial (a polynomial with leading coecient $1$) of degree $4$, such that $P(2^n+1) =8^n + 1$ when $n = 1, 2, 3, 4$. Find the value of $P(1)$. [b]p21[/b]. PEAcock and Zombie Hen Hao are at the starting point of a circular track, and start running in the same direction at the same time. PEAcock runs at a constant speed that is $2018$ times faster than Zombie Hen Hao's constant speed. At some point in time, Farmer James takes a photograph of his two favorite chickens, and he notes that they are at different points along the track. Later on, Farmer James takes a second photograph, and to his amazement, PEAcock and Zombie Hen Hao have now swapped locations from the first photograph! How many distinct possibilities are there for PEAcock and Zombie Hen Hao's positions in Farmer James's first photograph? (Assume PEAcock and Zombie Hen Hao have negligible size.) [u]Round 8[/u] [b]p22.[/b] How many ways are there to scramble the letters in $EGGSEATER$ such that no two consecutive letters are the same? [b]p23.[/b] Let $JAMES$ be a regular pentagon. Let $X$ be on segment $JA$ such that $\frac{JX}{XA} = \frac{XA}{JA}$ . There exists a unique point $P$ on segment $AE$ such that $XM = XP$. Find the ratio $\frac{AE}{PE}$ . [b]p24.[/b] Find the minimum value of the function $$f(x) = \left|x - \frac{1}{x} \right|+ \left|x - \frac{2}{x} \right| + \left|x - \frac{3}{x} \right|+... + \left|x - \frac{9}{x} \right|+ \left|x - \frac{10}{x} \right|$$ over all nonzero real numbers $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949191p26406082]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

1997 All-Russian Olympiad Regional Round, 11.8

For which $a$, there is a function $f: R \to R$, different from a constant, such that $$f(a(x + y)) = f(x) + f(y) ?$$

2018 Baltic Way, 20

Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.

2013 F = Ma, 24

Tags:
A man with mass $m$ jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the Bungee cord is four times the weight of the man. Find the maximum extension of the bungee cord $h$. $\textbf{(A) } h = \frac{1}{2}H \\ \\ \textbf{(B) } h = \frac{1}{4}H\\ \\ \textbf{(C) } h = \frac{1}{5}H\\ \\ \textbf{(D) } h = \frac{2}{5}H\\ \\ \textbf{(E) } h = \frac{1}{8}H$

2011 Bundeswettbewerb Mathematik, 1

Ten bowls are in a circle. They will go clockwise - starting somewhere filled with $1, 2, 3, ..., 9$ or $10$ marbles. You can have two choices in every move . Add a marble to neighboring shells or from two neighboring shells - if both of them are not empty - remove one marble each. Can you achieve that after finally many moves in each bowl exactly $2011$ marbles lying?

2011 Germany Team Selection Test, 2

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Brazil L2 Finals (OBM) - geometry, 2002.1

Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$.

2015 China Team Selection Test, 3

For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.

2009 Kyrgyzstan National Olympiad, 7

Tags: inequalities
Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.

2021 Baltic Way, 10

John has a string of paper where $n$ real numbers $a_i \in [0, 1]$, for all $i \in \{1, \ldots, n\}$, are written in a row. Show that for any given $k < n$, he can cut the string of paper into non-empty $k$ pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than $1$.

2025 JBMO TST - Turkey, 7

Tags: pentagon , ratio , geometry
$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?

2018 China Northern MO, 6

For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$ Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$

2013 Philippine MO, 5

Tags: inequalities
Let $r$ and $s$ be positive real numbers such that $(r+s-rs)(r+s+rs)=rs$. Find the minimum value of $r+s-rs$ and $r+s+rs$

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: inradius , geometry
The radius of the circle inscribed in triangle $ABC$ is equal to $r$. This circle is tangent to $BC$ at point $M$ and divides the segment $AM$ in ratio $k$ (starting from vertex $A$). Find the sum of the radii of the circles inscribed in triangles $AMB$ and $AMC$.

2016 District Olympiad, 3

Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$

2016 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt
A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.

2016 CMIMC, 3

Tags: geometry
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?

2020 HMNT (HMMO), 10

Sean enters a classroom in the Memorial Hall and sees a $1$ followed by $2020$ $0$'s on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\frac{n-1}{n}$. (In particular, the $1$ is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-$3$ number. (For example, if the remaining number on the board is $1000$, then its value is $27$.)

1991 Irish Math Olympiad, 2

Tags: inequalities
Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, \quad n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$. Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$.

2006 MOP Homework, 4

Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.