Olympiad problems

2019 Danube Mathematical Competition, 3

Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $

1996 Putnam, 4

Tags: linear algebra , matrix , trigonometry

For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

2023 Harvard-MIT Mathematics Tournament, 12

Tags: guts

The number $770$ is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either $40$ or $41$ from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N = a\cdot 2^b$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100a+b.$

1971 IMO Longlists, 28

Tags: 3d geometry , tetrahedron , trigonometry , distance , geometry

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

2021 Romania National Olympiad, 2

Tags: floor function , algebra , function

Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by \[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\] Prove that the following are equivalent: [list=1] [*] $f$ is surjective; [*] $c=0$, $b<d$ and $0<a\le d$. [/list] [i]Tiberiu Trif[/i]

2020 BMT Fall, 14

Tags: algebra

Let $B, M$, and $T$ be the three roots of the equation $x^3 + 20x^2 -18x-19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?

2019 HMNT, 6

Tags: combinatorics

Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $23$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?

2015 Geolympiad Summer, 5.

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Let $ABC$ be a triangle and $P$ be in its interior. Let $Q$ be the isogonal conjugate of $P$. Show that $BCPQ$ is cyclic if and only if $AP=AQ$.

1995 Abels Math Contest (Norwegian MO), 4

Tags: inequalities , algebra

Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

2018 IMO Shortlist, N1

Tags: number theory , arithmetic function , number of divisors , divisor

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

1998 USAMTS Problems, 2

Tags: greatest common divisor

Prove that there are inﬁnitely many ordered triples of positive integers $(a,b,c)$ such that the greatest common divisor of $a,b,$ and $c$ is $1$, and the sum $a^2b^2+b^2c^2+c^2a^2$ is the square of an integer.

2017 ASDAN Math Tournament, 18

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Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.

2012 HMNT, 9

Tags: geometry

Triangle $ABC$ satisfies $\angle B > \angle C$. Let $M$ be the midpoint of $BC$, and let the perpendicular bisector of $BC$ meet the circumcircle of $\vartriangle ABC$ at a point $D$ such that points $A$, $D$, $C$, and $B$ appear on the circle in that order. Given that $\angle ADM = 68^o$ and $\angle DAC = 64^o$ , find $\angle B$.

1990 Swedish Mathematical Competition, 2

Tags: sum , geometric inequalities , geometry , collinear

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1951 AMC 12/AHSME, 15

Tags: calculus , integration

The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

2025 India STEMS Category C, 4

Tags: function , calculus

Does there exist a function $f:[0,1]\rightarrow (0,\infty)$ such that [list] [*]$f$ is differentiable on $[0,1]$ [*] It's derivative $f'$ is continuous on $[0,1]$. [*] $(f'(x))^3-x^{\frac{1}{3}}>6(1-f(x)^{\frac{1}{5}})$ for all $x\in [0,1]$. [*] $f(1)=1$ [/list] [i]Proposed by Medhansh Tripathi[/i]

2006 Estonia National Olympiad, 4

Tags: trigonometry , geometry

Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.

2023 Germany Team Selection Test, 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.

2003 AMC 10, 14

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Given that $ 3^8\cdot5^2 \equal{} a^b$, where both $ a$ and $ b$ are positive integers, find the smallest possible value for $ a \plus{} b$. $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 351 \qquad \textbf{(D)}\ 407 \qquad \textbf{(E)}\ 900$

2016 Ecuador Juniors, 3

Tags: geometry

Let $P_1P_2 . . . P_{2016 }$ be a cyclic polygon of $2016$ sides. Let $K$ be a point inside the polygon and let $M$ be the midpoint of the segment $P_{1000}P_{2000}$. Knowing that $KP_1 = KP_{2011} = 2016$ and $KM$ is perpendicular to $P_{1000}P_{2000}$, find the length of segment $KP_{2016}$.

2010 Silk Road, 3

Tags: inequalities

For positive real numbers $a, b, c, d,$ satisfying the following conditions: $a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$

1984 Austrian-Polish Competition, 7

Tags: linear algebra , matrix

A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.

1986 IMO Longlists, 43

Tags: probability , combinatorics , game , game strategy , counting

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

2016 ASDAN Math Tournament, 5

Tags:

Find $$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$

2011 Croatia Team Selection Test, 2

Tags: analytic geometry , combinatorics

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$