Olympiad problems

2021 JHMT HS, 4

Tags: combinatorics

For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute \[ \sum_{n=46}^{86} f(n). \]

2018 Nepal National Olympiad, 3b

Tags: geometry

[b] Problem Section #3 NOTE: Neglect that HF and CD.

2024 CIIM, 1

Tags: integration , function , recurrence relation , calculus , sequence

Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds: \[ f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1. \] Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\ [b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.

2022 AMC 10, 13

Tags: prime , difference of cubes

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes? $\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$

2008 Brazil National Olympiad, 3

Tags: trigonometry , inequalities , calculus

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

2022 Kyiv City MO Round 2, Problem 2

Tags: combinatorics , coloring

$2022$ points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions: 1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white) 2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them Is it possible to achieve a configuration where each point has a color opposite to its initial color with these operations? [i](Proposed by Oleksii Masalitin)[/i]

2010 Math Prize For Girls Problems, 5

Tags: prime factorization , number theory

Find the smallest two-digit positive integer that is a divisor of 201020112012.

PEN O Problems, 36

Tags: induction , inequalities

Let a and b be non-negative integers such that $ab \ge c^{2}$ where $c$ is an integer. Prove that there is a positive integer n and integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$, $y_{1}$, $y_{2}$, $\cdots$, $y_{n}$ such that \[{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c\]

2012 AMC 10, 1

Tags:

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

2019 India IMO Training Camp, P3

Tags: algebra

Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

2014 Belarus Team Selection Test, 1

Tags: geometry , circles , concyclic , collinear

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

2013 HMNT, 10

Tags: combinatorics , algebra

Let $\omega= \cos \frac{2\pi}{727} + i \sin \frac{2\pi}{727}$. The imaginary part of the complex number $$\prod^{13}_{k=8} \left(1 + \omega^{3^{k-1}}+ \omega^{2\cdot 3^{k-1}}\right)$$ is equal to $\sin a$ for some angle $a$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ , inclusive. Find $a$.

2004 Bulgaria Team Selection Test, 2

Tags: ramsey theory , graph theory , combinatorics

The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.

2023 Assam Mathematics Olympiad, 15

Tags:

Let $f(x)$ be a polynomial of degree $3$ with real coefficients satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.

2007 Moldova Team Selection Test, 1

Tags: algebra , geometry , 3d geometry , factorization , sum of cubes , number theory

Find the least positive integers $m,k$ such that a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube. b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu

2018 China National Olympiad, 4

Tags: geometry

$ABCD$ is a cyclic quadrilateral whose diagonals intersect at $P$. The circumcircle of $\triangle APD$ meets segment $AB$ at points $A$ and $E$. The circumcircle of $\triangle BPC$ meets segment $AB$ at points $B$ and $F$. Let $I$ and $J$ be the incenters of $\triangle ADE$ and $\triangle BCF$, respectively. Segments $IJ$ and $AC$ meet at $K$. Prove that the points $A,I,K,E$ are cyclic.

2022 CMIMC, 15

Tags: team

Let $ABC$ be a triangle with $AB = 5, BC = 13,$ and $AC = 12$. Let $D$ be a point on minor arc $AC$ of the circumcircle of $ABC$ (endpoints excluded) and $P$ on $\overline{BC}$. Let $B_1, C_1$ be the feet of perpendiculars from $P$ onto $CD, AB$ respectively and let $BB_1, CC_1$ hit $(ABC)$ again at $B_2, C_2$ respectively. Suppose that $D$ is chosen uniformly at random and $AD, BC, B_2C_2$ concur at a single point. Compute the expected value of $BP/PC$. [i]Proposed by Grant Yu[/i]

2000 National High School Mathematics League, 1

Tags:

If $A=\{x|\sqrt{x-2}\leq0\},B=\{x|10^{x^2-2}=10^{x}\}$, then $A\cap(\mathbb{R}\backslash B)$ is $\text{(A)}\{2\}\qquad\text{(B)}\{-1\}\qquad\text{(C)}\{x|x\leq2\}\qquad\text{(D)}\varnothing$

2003 Moldova Team Selection Test, 3

Tags: geometry

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

2017 F = ma, 17

Tags: kinematics

17) An object is thrown directly downward from the top of a 180-meter-tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof? A) 15 m/s B) 25 m/s C) 35 m/s D) 55 m/s E) insufficient information

1994 Miklós Schweitzer, 1

Tags: ordered set

An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$. Is it possible to order the real numbers so it becomes mean-free? related: [url]https://www.youtube.com/watch?v=ppaXUxsEjMQ[/url]

2017 Bulgaria JBMO TST, 1

Tags: number theory

Find all positive integers $ a, b, c, d $ so that $ a^2+b^2+c^2+d^2=13 \cdot 4^n $

2022 Saudi Arabia BMO + EGMO TST, 1.3

Tags: number theory

Let $p$ be a prime number and let $m, n$ be integers greater than $1$ such that $n | m^{p(n-1)} - 1$. Prove that $gcd(m^{n-1} - 1, n) > 1$.

2019 Harvard-MIT Mathematics Tournament, 4

Tags: geometry , hmmt

Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

2019 Dutch IMO TST, 4

Tags: geometry , cyclic quadrilateral , circumcircle , parallel

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic