Olympiad problems

2016 Taiwan TST Round 1, 2

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2018 Putnam, B3

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Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.

2023 Belarus Team Selection Test, 3.1

Tags: algebra

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2014 Balkan MO Shortlist, N5

Tags: number theory

$\boxed{N5}$Let $a,b,c,p,q,r$ be positive integers such that $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.$ Prove that $a=b=c$ or $p=q=r.$

2015 Singapore Senior Math Olympiad, 2

Tags: sequence , circles , algebra , combinatorics

There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

2013 AIME Problems, 8

Tags: logarithm , domain , function , modular arithmetic , algebra , trigonometry

The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

2003 IberoAmerican, 3

Tags: number theory , quadratic , induction , modular arithmetic

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

2007 F = Ma, 21

Tags: rotation , geometric transformation , 3d geometry , sphere , geometry

If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius? $ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $

2014 Harvard-MIT Mathematics Tournament, 9

Tags: dilation , incenter , inradius , geometric transformation , trigonometry , geometry

Two circles are said to be [i]orthogonal[/i] if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_1$ and $\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\omega_3$ with radius $2\sqrt2$ passes through $P$ and is orthogonal to both $\omega_1$ and $\omega_2$. A fourth circle $\omega_4$, orthogonal to $\omega_3$, is externally tangent to $\omega_1$ and $\omega_2$. Compute the radius of $\omega_4$.

2018 VJIMC, 2

Tags: prime number , number theory

Find all prime numbers $p$ such that $p^3$ divides the determinant \[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]

2020 Harvard-MIT Mathematics Tournament, 1

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Let $n$ be a positive integer. Define a sequence by $a_0 = 1$, $a_{2i+1} = a_i$, and $a_{2i+2} = a_i + a_{i+1}$ for each $i \ge 0$. Determine, with proof, the value of $a_0 + a_1 + a_2 + \dots + a_{2^n-1}$. [i]Proposed by Kevin Ren.[/i]

2006 Bulgaria National Olympiad, 1

Tags: combinatorics

Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$. [i]Aleksandar Ivanov[/i]

2016 Mathematical Talent Reward Programme, MCQ: P 12

Tags: continuity , function

Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are [list=1] [*] 0 [*] 1 [*] 2 [*] More than 2 [/list]

JOM 2023, 3

Tags: geometry

Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$. [i]Proposed by Wong Jer Ren[/i]

2012-2013 SDML (Middle School), 14

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Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to the three trick-or-treaters given these restrictions? $\text{(A) }91\qquad\text{(B) }90\qquad\text{(C) }81\qquad\text{(D) }72\qquad\text{(E) }70$

2007 Pan African, 3

Tags: geometry

An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?

2024 Miklos Schweitzer, 11

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An urn initially contains one red ball and one blue ball. In each step, we choose a uniform random ball from the urn. If it is red, then another red ball and another blue ball are placed in the urn. And when we choose a blue ball for the $k$-th time, we put a blue ball and $2k + 1$ red balls in the urn. (The chosen balls are not removed; they remain in the urn.) Let $G_n$ denote the number of red balls in the urn after $n$ steps. Prove that there exist constants $0 < c, \alpha < \infty$ such that $\frac{G_n}{n^\alpha} \to c$ almost surely.

2021 MOAA, 6

Tags: speed

Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$. [i]Proposed by Nathan Xiong[/i]

2009 Postal Coaching, 3

Tags: polygon , cyclic , geometric inequalities , inequalities

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

2013 Saudi Arabia BMO TST, 6

Tags: algebra , inequalities

Let $a, b,c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that $$a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3$$

2019 Ramnicean Hope, 2

Tags: real analysis , function graphing , infimum

Calculate $ \inf_{x> 0} \sqrt{(1+x)^2+4/x} . $ [i]Constantin Rusu[/i] and [i]Mihai Neagu[/i]

2003 VJIMC, Problem 4

Tags: function , calculus , integration , inequalities

Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that $$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$

2013 BMT Spring, 5

Tags: geometry , circles

Circle $C_1$ has center $O$ and radius $OA$, and circle $C_2$ has diameter $OA$. $AB$ is a chord of circle $C_1$ and $BD$ may be constructed with $D$ on $OA$ such that $BD$ and $OA$ are perpendicular. Let $C$ be the point where $C_2$ and $BD$ intersect. If $AC = 1$, find $AB$.

2021 Thailand Mathematical Olympiad, 3

Tags: inequalities

Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of $$a^abc + b^bca + c^cab.$$

1992 National High School Mathematics League, 2

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Define set $S_n=\{1,2,\cdots,n\}$. $X$ is a subset of $S_n$. We call sum of all numbers in $X$ [i]capacity[/i] ([i]capacity[/i] of empty set is $0$). If [i]capacity[/i] of $X$ is odd/even, then we call it [i]odd/even subset[/i]. [b](a)[/b] Prove that the number of [i]odd subsets[/i] and [i]even subsets[/i] of $S_n$ are the same. [b](b)[/b] Prove that the sum of [i]capacity[/i] of all [i]odd subsets[/i] and [i]even subsets[/i] are the same when $n\geq3$. [b](c)[/b] Calculate the sum of [i]capacity[/i] of all [i]odd subsets[/i] when $n\geq3$.