Olympiad problems

1957 Moscow Mathematical Olympiad, 349

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

2014-2015 SDML (High School), 4

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Evaluate $$1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.$$ $\text{(A) }\frac{\sqrt{3}}{2}\qquad\text{(B) }\frac{1+\sqrt{5}}{2}\qquad\text{(C) }\frac{2+\sqrt{3}}{2}\qquad\text{(D) }\frac{3+\sqrt{5}}{2}\qquad\text{(E) }\frac{3+\sqrt{13}}{2}$

2002 AMC 8, 2

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How many different combinations of $5$ bills and $2$ bills can be used to make a total of $17$? Order does not matter in this problem. $ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $

1999 Baltic Way, 2

Tags: geometry , 3d geometry , number theory

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

2006 Switzerland Team Selection Test, 2

Tags: algebra , geometry

Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

Kyiv City MO Seniors Round2 2010+ geometry, 2011.10.4

Tags: geometry , perpendicular , parallel , tangent circle

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

2009 USAMTS Problems, 4

Tags: geometric transformation , reflection , geometry

Let $ABCDEF$ be a convex hexagon, such that $FA = AB$, $BC = CD$, $DE = EF$, and $\angle FAB = 2\angle EAC$. Suppose that the area of $ABC$ is $25$, the area of $CDE$ is $10$, the area of $EF A$ is $25$, and the area of $ACE$ is $x$. Find, with proof, all possible values of $x$.

2021 BMT, T4

Tags: complex number , algebra

Let $z_1$, $z_2$, and $z_3$ be the complex roots of the equation $(2z -3\overline{z})^3 = 54i+54$. Compute the area of the triangle formed by $z_1$, $z_2$, and $z_3$ when plotted in the complex plane.

2000 Harvard-MIT Mathematics Tournament, 9

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$\frac{a}{c}=\frac{b}{d}=\frac{3}{4}$, $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$. Find $ac+bd-ad-bc$.

2017 India PRMO, 9

Tags: combinatorics

There are five cities $A,B,C,D,E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes $A \to B \to C \to A$ and $A\to C \to B \to A$ are different.)

1977 Dutch Mathematical Olympiad, 3

Tags: number theory , divide , divisible , multiple

From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

2009 Germany Team Selection Test, 3

Tags: geometry , circumcircle , reflection

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

2005 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , function , derivative , real analysis

Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

1969 IMO Shortlist, 14

Tags: quadratic , algebra , inequality , root , polynomial

$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

2016 APMO, 1

Tags: circumcircle , functional equation , right triangle , reflection , geometry

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2014 Paraguay Mathematical Olympiad, 3

Tags: algebra

Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?

2008 Germany Team Selection Test, 2

Tags: combinatorics , modular arithmetic , counting , triplets

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

2020 Azerbaijan IMO TST, 3

Tags: algebra , product , determinant

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

1994 Turkey MO (2nd round), 2

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$. Point $E$ is taken on segment $DA$ such that $BD=2DE$. The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$. Prove that \[ \frac{AC\cdot BD}{AB\cdot FC}=2.\]

1988 All Soviet Union Mathematical Olympiad, 484

Tags: trigonometry , system of equations , minimum , algebra , sum

What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?

2000 National High School Mathematics League, 4

Tags: geometric series , arithmetic sequence

Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$? $\text{(A)}$ It has no real roots. $\text{(B)}$ It has two equal real roots. $\text{(C)}$ It has two different real roots, and their product is positive. $\text{(D)}$ It has two different real roots, and their product is negative.

2023 China Girls Math Olympiad, 4

Tags: geometry

Let $ABCD$ be an inscribed quadrilateral of some circle $\omega$ with $AC\ \bot \ BD$. Define $E$ to be the intersection of segments $AC$ and $BD$. Let $F$ be some point on segment $AD$ and define $P$ to be the intersection point of half-line $FE$ and $\omega$. Let $Q$ be a point on segment $PE$ such that $PQ\cdot PF = PE^2$. Let $R$ be a point on $BC$ such that $QR\ \bot \ AD$. Prove that $PR=QR$.

2003 AMC 12-AHSME, 4

Tags: geometry

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

2008 Indonesia TST, 3

Tags: gcd , greatest common divisor , number theory

Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).

2004 Kazakhstan National Olympiad, 5

Tags: polynomial , algebra

Let $ P (x) $ be a polynomial with real coefficients such that $ P (x)> 0 $ for all $ x \geq 0 $. Prove that there is a positive integer $ n $ such that $ (1 + x) ^ n P (x) $ polynomial with nonnegative coefficients.