Olympiad problems

Denmark (Mohr) - geometry, 2012.1

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

2004 Germany Team Selection Test, 2

Tags: geometry , locus problems , geometric inequalities

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

2021 CCA Math Bonanza, I4

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Given that nonzero real numbers $x$ and $y$ satisfy $x+\frac{1}{y}=3$ and $y+\frac{1}{x}=4$, what is $xy+\frac{1}{xy}$? [i]2021 CCA Math Bonanza Individual Round #4[/i]

Indonesia MO Shortlist - geometry, g10

Tags: geometry , incenter , circles

Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.

2016 AMC 12/AHSME, 1

Tags: factorial , fraction

What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2010 South africa National Olympiad, 2

Tags: trigonometry , geometry

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

2004 Cono Sur Olympiad, 5

Tags: combinatorial geometry , combinatorics

Using cardboard equilateral triangles of side length $1$, an equilateral triangle of side length $2^{2004}$ is formed. An equilateral triangle of side $1$ whose center coincides with the center of the large triangle is removed. Determine if it is possible to completely cover the remaining surface, without overlaps or holes, using only pieces in the shape of an isosceles trapezoid, each of which is created by joining three equilateral triangles of side $1$.

1985 IMO Longlists, 71

Tags: inequalities , number theory

For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.

2010 Romania Team Selection Test, 2

Tags: geometry , circumcircle , incenter

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$. [i]Cezar Lupu & Vlad Matei[/i]

2019 Danube Mathematical Competition, 3

Tags: number theory , combinatorics , sequence

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]