Olympiad problems

2020 CCA Math Bonanza, I7

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Define the binary operation $a\Delta b=ab+a-1$. Compute \[ 10 \Delta(10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10)))))))) \] where $10$ is written $10$ times. [i]2020 CCA Math Bonanza Individual Round #7[/i]

1983 National High School Mathematics League, 5

Tags: function

$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then $\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$

2010 Hanoi Open Mathematics Competitions, 5

Tags: combinatorics , coloring

Each box in a $2x2$ table can be colored black or white. How many different colorings of the table are there? (A): $4$, (B): $8$, (C): $16$, (D): $32$, (E) None of the above.

2016 Czech And Slovak Olympiad III A, 4

Tags: inequalities , maximum value , maximum , algebra

For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$. Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value

2007 ITest, 38

Tags: geometry , 3d geometry , floor function , inequalities , logarithm

Find the largest positive integer that is equal to the cube of the sum of its digits.

1954 AMC 12/AHSME, 11

Tags: ratio

A merchant placed on display some dresses, each with a marked price. He then posted a sign “$ \frac{1}{3}$ off on these dresses.” The cost of the dresses was $ \frac{3}{4}$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was: $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{3}{4}$

2002 Hong kong National Olympiad, 3

Tags: inequalities

Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.

2001 AIME Problems, 3

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Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0,$ given that there are no multiple roots.

2006 District Olympiad, 2

Tags: matrix , inequalities , algebra , polynomial , abstract algebra , linear algebra

Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$. a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$. b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]

2009 Italy TST, 1

Tags: vector , linear algebra , matrix , combinatorics

Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations? i)$k$ is a prime number greater than $2$; ii) $k$ is odd; iii) $k$ is even.

2017 Thailand TSTST, 4

Tags: cell , combinatorics , coloring , table , construction

The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?

2020 China Girls Math Olympiad, 3

Tags: combinatorics

There are $3$ classes with $n$ students in each class, and the heights of all $3n$ students are pairwise distinct. Partition the students into groups of $3$ such that in each group, there is one student from each class. In each group, call the tallest student the [i]tall guy[/i]. Suppose that for any partition of the students, there are at least 10 tall guys in each class, prove that the minimum value of $n$ is $40$.

2015 Saint Petersburg Mathematical Olympiad, 2

Tags: number theory

$a,b>1$ - are naturals, and $a^2+b,a+b^2$ are primes. Prove $(ab+1,a+b)=1$

2018 ISI Entrance Examination, 4

Tags: calculus

Let $f:(0,\infty)\to\mathbb{R}$ be a continuous function such that for all $x\in(0,\infty)$, $$f(2x)=f(x)$$ Show that the function $g$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$ is a constant function.

2021 South East Mathematical Olympiad, 3

Tags: number theory , algebra

Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence. Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients. If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.

1963 Swedish Mathematical Competition., 3

Tags: number theory , remainder

What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?

1980 Dutch Mathematical Olympiad, 2

Tags: number theory , divisor

Find the product of all divisors of $1980^n$, $n \ge 1$.

2019 Peru IMO TST, 3

Tags: circumcircle , incenter , mixtilinear incircle , geometry

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

1988 AIME Problems, 11

Tags: analytic geometry , graphing lines , slope , geometry , geometric transformation , ratio , complex number

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \[ \sum_{k = 1}^n (z_k - w_k) = 0. \] For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.

1991 Arnold's Trivium, 27

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Sketch the images of the solutions of the equation \[\ddot{x}=F(x)-k\dot{x},\; F=-dU/dx\] in the $(x,E)$-plane, where $E=\dot{x}^2/2+U(x)$, near non-degenerate critical points of the potential $U$.

VMEO III 2006, 11.2

Tags: geometry

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

1974 AMC 12/AHSME, 8

Tags: modular arithmetic

What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$

2022/2023 Tournament of Towns, P1

Tags: combinatorics , geometry

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).

1997 Moscow Mathematical Olympiad, 4

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Given real numbers $a_1\leq{a_2}\leq{a_3}$ and $b_1\leq{b_2}\leq{b_3}$ such that $$a_1+a_2+a_3=b_1+b_2+b_3,$$ $$a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.$$ Prove that if $a_1\leq{b_1},$ then $a_3\leq{b_3}$

2005 Today's Calculation Of Integral, 84

Tags: calculus , integration , limit , trigonometry , calculus computations

Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]