Olympiad problems

1984 Swedish Mathematical Competition, 6

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

1973 Kurschak Competition, 3

Tags: combinatorics , combinatorial geometry

$n > 4$ planes are in general position (so every $3$ planes have just one common point, and no point belongs to more than $3$ planes). Show that there are at least $\frac{2n-3}{ 4}$ tetrahedra among the regions formed by the planes.

2001 ITAMO, 1

Tags: geometry

A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.

2009 Germany Team Selection Test, 1

Tags: prime number , number theory

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2024 USAJMO, 3

Tags: awesomeness_in_a_bun

Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$. [i]Proposed by John Berman[/i]

2016 Middle European Mathematical Olympiad, 5

Tags: circumcircle , law of cosines , geometry

Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.

2011 NIMO Problems, 11

Tags: number theory , least common multiple , greatest common divisor

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

1990 Romania Team Selection Test, 3

Tags: polynomial , algebra , functional , functional equation

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2004 Purple Comet Problems, 3

Tags: algebra , polynomial

How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

2022 Putnam, A5

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Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

2018 Purple Comet Problems, 1

Tags: algebra

Find the positive integer $n$ such that $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 + \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$ .

1952 AMC 12/AHSME, 28

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In the table shown, the formula relating $ x$ and $ y$ is: \[ \begin{tabular}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 7 & 13 & 21 & 31 \\ \hline \end{tabular} \]$ \textbf{(A)}\ y \equal{} 4x \minus{} 1 \qquad\textbf{(B)}\ y \equal{} x^3 \minus{} x^2 \plus{} x \plus{} 2 \qquad\textbf{(C)}\ y \equal{} x^2 \plus{} x \plus{} 1$ $ \textbf{(D)}\ y \equal{} (x^2 \plus{} x \plus{} 1)(x \minus{} 1) \qquad\textbf{(E)}\ \text{none of these}$

2001 Putnam, 1

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Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.

2017 Serbia National Math Olympiad, 3

Tags: geometry , excircle

Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.

2023 CMIMC Team, 13

Tags: team

Suppose that the sequence of real numbers $a_1,a_2,\ldots$ satisfies $a_1 = - \sqrt{1}, a_2 = \sqrt{2}$, and for all $k > 1$, \[ \frac{a_{k+1}+a_{k-1}}{a_k} = \frac{\sqrt{3} + \sqrt{1}}{\sqrt{2}}. \] Find $a_{2023}$. [i]Proposed by Kevin You[/i]

1997 Akdeniz University MO, 2

Tags: inequalities , algebra

Let $x,y,z,t$ be real numbers such that, $1 \leq x \leq y \leq z \leq t \leq 100$. Find minimum value of $$\frac{x}{y}+\frac{z}{t}$$

2006 China National Olympiad, 1

Tags: induction , inequalities

Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$. Prove that \[ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right). \]

Geometry Mathley 2011-12, 16.4

Tags: geometry , orthocenter , incenter , circumcenter

A triangle $ABC$ is inscribed in the circle $(O)$, and has incircle $(I)$. The circles with diameter $IA$ meets $(O)$ at $A_1$ distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Line $B_1C_1$ meets $BC$ at $A_2$, and points $B_2,C_2$ are defined in the same manner. Prove that $O$ is the orthocenter of triangle $A_2B_2C_2$. Trần Minh Ngọc

2015 Spain Mathematical Olympiad, 1

Tags: polyhedron , graph coloring , combinatorics

All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?

OMMC POTM, 2023 6

Tags: combinatorics

Choose a permutation of$ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$. [i]Proposed by Evan Chang (squareman), USA[/i]

1997 AMC 8, 3

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Which of the following numbers is the largest? $\textbf{(A)}\ 0.97 \qquad \textbf{(B)}\ 0.979 \qquad \textbf{(C)}\ 0.9709 \qquad \textbf{(D)}\ 0.907 \qquad \textbf{(E)}\ 0.9089$

2016 CCA Math Bonanza, I8

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Let $f(x) = x^2 + x + 1$. Determine the ordered pair $(p,q)$ of primes satisfying $f(p) = f(q) + 242$. [i]2016 CCA Math Bonanza #8[/i]

2022 Estonia Team Selection Test, 2

Tags: number theory , first digit

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

2016 India Regional Mathematical Olympiad, 5

Tags: number theory , sum of squares

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

2005 National Olympiad First Round, 31

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If the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real roots, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, at most how many distinct real roots can the equation $f(x)g(x) = 0$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $