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: 15925

2023 Indonesia TST, 1

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 $a_{2022}\leq 1$.

2021 Princeton University Math Competition, B2

Tags: algebra
Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$.

2019 Romania EGMO TST, P3

Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$

1995 Czech and Slovak Match, 4

For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$.

2013 ELMO Problems, 6

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2004 Indonesia Juniors, day 2

p1. A regular $6$-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is $ 12$? p2. Given two positive real numbers $x$ and $y$ with $xy = 1$. Determine the minimum value of $\frac{1}{x^4}+\frac{1}{4y^4}.$ p3. Known a square network which is continuous and arranged in forming corners as in the following picture. Determine the value of the angle marked with the letter $x$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/aee36501b00c4aaeacd398f184574bd66ac899.png[/img] p4. Find the smallest natural number $n$ such that the sum of the measures of the angles of the $n$-gon, with $n > 6$ is less than $n^2$ degrees. p5. There are a few magic cards. By stating on which card a number is there, without looking at the card at all, someone can precisely guess the number. If the number is on Card $A$ and $B$, then the number in question is $1 + 2$ (sum of corner number top left) cards $A$ and $B$. If the numbers are in $A$, $B$, and $C$, the number what is meant is $1 + 2 + 4$ or equal to $7$ (which is obtained by adding the numbers in the upper left corner of each card $A$,$B$, and $C$). [img]https://cdn.artofproblemsolving.com/attachments/e/5/9e80d4f3bba36a999c819c28c417792fbeff18.png[/img] a. How can this be explained? b. Suppose we are going to make cards containing numbers from $1$ to with $15$ based on the rules above. Try making the cards. [hide=original wording for p5, as the wording isn't that clear]Ada suatu kartu ajaib. Dengan menyebutkan di kartu yang mana suatu bilan gan berada, tanpa melihat kartu sama sekali, seseorang dengan tepat bisa menebak bilangan yang dimaksud. Kalau bilangan tersebut ada pada Kartu A dan B, maka bilangan yang dimaksud adalah 1 + 2 (jumlah bilangan pojok kiri atas) kartu A dan B. Kalau bilangan tersebut ada di A, B, dan C, bilangan yang dimaksud adalah 1 + 2 + 4 atau sama dengan 7 (yang diperoleh dengan menambahkan bilangan-bilangan di pojok kiri atas masing-masing kartu A, B, dan C) a. Bagaimana hal ini bisa dijelaskan? b. Andai kita akan membuat kartu-kartu yang memuat bilangan dari 1 sampai dengan 15 berdasarkan aturan di atas. Coba buatkan kartu-kartunya[/hide]

2016 Korea Winter Program Practice Test, 4

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

1968 IMO Shortlist, 11

Tags: equation , algebra , root
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

LMT Guts Rounds, 2019 S

[u]Round 1[/u] [b]p1.[/b] Alice has a pizza with eight slices. On each slice, she either adds only salt, only pepper, or leaves it plain. Determine how many ways there are for Alice to season her entire pizza. [b]p2.[/b] Call a number almost prime if it has exactly three divisors. Find the number of almost prime numbers less than $100$. [b]p3.[/b] Determine the maximum number of points of intersection between a circle and a regular pentagon. [u]Round 2[/u] [b]p4.[/b] Let $d(n)$ denote the number of positive integer divisors of $n$. Find $d(d(20^{18}))$. [b]p5.[/b] $20$ chubbles are equal to $19$ flubbles. $20$ flubbles are equal to $18$ bubbles. How many bubbles are $1000$ chubbles worth? [b]p6.[/b] Square $ABCD$ and equilateral triangle $EFG$ have equal area. Compute $\frac{AB}{EF}$ . [u]Round 3[/u] [b]p7.[/b] Find the minimumvalue of $y$ such that $y = x^2 -6x -9$ where x is a real number. [b]p8.[/b] I have $2$ pairs of red socks, $5$ pairs of white socks, and $7$ pairs of blue socks. If I randomly pull out one sock at a time without replacement, how many socks do I need to draw to guarantee that I have drawn $3$ pairs of socks of the same color? [b]p9. [/b]There are $23$ paths from my house to the school, $29$ paths from the school to the library, and $3$ paths fromthe library to town center. Additionally, there are $6$ paths directly from my house to the library. If I have to pass through the library to get to town center, howmany ways are there to travel from my house all the way to the town center? [u]Round 4[/u] [b]p10.[/b] A circle of radius $25$ and a circle of radius $4$ are externally tangent. A line is tangent to the circle of radius $25$ at $A$ and the circle of radius $4$ at $B$, where $A \ne B$. Compute the length of $AB$. [b]p11.[/b] A gambler spins two wheels, one numbered $1$ to $4$ and another numbered $1$ to $5$, and the amount of money he wins is the sum of the two numbers he spins in dollars. Determine the expected amount of money he will win. [b]p12.[/b] Find the remainder when $20^{19}$ is divided by $18$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2005 IMO Shortlist, 2

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

2013 Saudi Arabia BMO TST, 3

Tags: max , sum , product , algebra
Let $T$ be a real number satisfying the property: For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$. Determine the minimum value of $T$.

2021 Indonesia TST, A

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

2022 Iran MO (3rd Round), 1

Tags: function , algebra
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$ $$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$

1998 Mediterranean Mathematics Olympiad, 2

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

2018 CHMMC (Fall), Individual

[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines $y = x$ or $y = -x$. Both robots take the shortest possible path to $(a, b)$ and arrive at the same time. Find the ratio $\frac{a}{b}$ . [b]p2.[/b] Suppose $x + \frac{1}{x} + y + \frac{1}{y} = 12$ and $x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70$. Compute $x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}$. [b]p3.[/b] Find the largest non-negative integer $a$ such that $2^a$ divides $$3^{2^{2018}}+ 3.$$ [b]p4.[/b] Suppose $z$ and $w$ are complex numbers, and $|z| = |w| = z \overline{w}+\overline{z}w = 1$. Find the largest possible value of $Re(z + w)$, the real part of $z + w$. [b]p5.[/b] Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $B$ each take turns making moves, with $A$ making the first move. The last player able to make a move wins. Suppose that the three piles have $10$, $x$, and $30$ matches. Find the largest value of $x$ for which $A$ does not have a winning strategy. [b]p6.[/b] Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon with side length $1$. For $n = 1$,$...$, $6$, let $B_n$ be a point on the segment $A_nA_{n+1}$ chosen at random (where indices are taken mod $6$, so $A_7 = A_1$). Find the expected area of the hexagon $B_1B_2B_3B_4B_5B_6$. [b]p7.[/b] A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can the termite use to reach the surface of the octahedron? [b]p8.[/b] Let $$P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n$$ Find the number of roots $z$ of $P(x)$ with $|z| > 1$, counting multiplicity. [b]p9.[/b] How many times does $01101$ appear as a not necessarily contiguous substring of $0101010101010101$? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read $01101$?) [b]p10.[/b] A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define $$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$ (where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers. [b]p11.[/b] Let $O$ be a circle with chord $AB$. The perpendicular bisector to $AB$ is drawn, intersecting $O$ at points $C$ and $D$, and intersecting $AB$ at the midpoint $E$. Finally, a circle $O'$ with diameter $ED$ is drawn, and intersects the chord $AD$ at the point $F$. Given $EC = 12$, and $EF = 7$, compute the radius of $O$. [b]p12.[/b] Suppose $r$, $s$, $t$ are the roots of the polynomial $x^3 - 2x + 3$. Find $$\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.$$ [b]p13.[/b] Let $a_1$, $a_2$,..., $a_{14}$ be points chosen independently at random from the interval $[0, 1]$. For $k = 1$, $2$,$...$, $7$, let $I_k$ be the closed interval lying between $a_{2k-1}$ and $a_{2k}$ (from the smaller to the larger). What is the probability that the intersection of $I_1$, $I_2$,$...$, $I_7$ is nonempty? [b]p14.[/b] Consider all triangles $\vartriangle ABC$ with area $144\sqrt3$ such that $$\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.$$ Over all such triangles $ABC$, what is the smallest possible perimeter? [b]p15.[/b] Let $N$ be the number of sequences $(x_1,x_2,..., x_{2018})$ of elements of $\{1, 2,..., 2019\}$, not necessarily distinct, such that $x_1 + x_2 + ...+ x_{2018}$ is divisible by $2018$. Find the last three digits of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2024 Dutch IMO TST, 2

Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with \[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\] for all $x,y,z \in \mathbb{R}_{\ge 0}$.

1956 Polish MO Finals, 2

Tags: algebra
Prove that if $$ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a + b + c}$$ and $ n $ is any odd natural number, then $$ \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n} =\frac{1}{a^n + b^n + c^n}$$

2009 Brazil Team Selection Test, 2

Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that $$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$

2022 CMIMC, 2.8 1.4

Let $z$ be a complex number that satisfies the equation \[\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.\] Over all possible values of $z$, find the sum of the values of \[\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.\] [i]Proposed by Justin Hsieh[/i]

2022 Mid-Michigan MO, 5-6

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2013 USAMTS Problems, 2

Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.

2006 Bulgaria Team Selection Test, 1

Tags: algebra , induction
Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

1984 Vietnam National Olympiad, 1

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

1996 All-Russian Olympiad Regional Round, 11.5

Tags: algebra
Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?

Mathematical Minds 2023, P7

Tags: sequence , algebra
Does there exist an increasing sequence of positive integers for which any large enough integer can be expressed uniquely as the sum of two (possibly equal) terms of the sequence? [i]Proposed by Vlad Spătaru and David Anghel[/i]