Olympiad problems

2016 Macedonia National Olympiad, Problem 3

Tags: number theory

Solve the equation in the set of natural numbers $xyz+yzt+xzt+xyt = xyzt + 3$

2000 AMC 10, 10

Tags: geometry , inequalities , triangle inequality

The sides of a triangle with positive area have lengths $ 4$, $ 6$, and $ x$. The sides of a second triangle with positive area have lengths $ 4$, $ 6$, and $ y$. What is the smallest positive number that is [b]not[/b] a possible value of $ |x \minus{} y|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$

LMT Theme Rounds, 2

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Mater is confused and starts going around the track in the wrong direction. He can go around 7 times in an hour. Lightning and Chick start in the same place at Mater and at the same time, both going the correct direction. Lightning can go around 91 times per hour, while Chick can go around 84 times per hour. When Lightning passes Chick for the third time, how many times will he have passed Mater (if Lightning is passing Mater just as he passes Chick for the third time, count this as passing Mater)? [i]Proposed by Matthew Weiss

1968 Czech and Slovak Olympiad III A, 4

Tags: geometry , 3d geometry , sphere , circumcircle

Four different points $A,B,C,D$ are given in space such that $AC\perp BD,AD\perp BC.$ Show there is a sphere containing midpoits of all 7 segments $AB,AC,AD,BC,BD,CD.$

1981 AMC 12/AHSME, 4

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If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is $\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 44 \qquad \text{(E)}\ 52$

2000 All-Russian Olympiad Regional Round, 10.3

Tags: ratio , geometry , parallelogram

Given a parallelogram $ABCD$ with angle $A$ equal to $60^o$. Point $O$ is the the center of a circle circumscribed around triangle $ABD$. Line $AO$ intersects the bisector of the exterior angle $C$ at point $K$. Find the ratio $AO/OK$.

2007 National Olympiad First Round, 20

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The $9$ consequtive sections of a paper strip are colored either red or white. If no two consequtive sections are white, in how many ways can this coloring be made? $ \textbf{(A)}\ 34 \qquad\textbf{(B)}\ 89 \qquad\textbf{(C)}\ 128 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 360 $

1974 AMC 12/AHSME, 13

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Which of the following is equivalent to "If P is true, then Q is false."? $ \textbf{(A)}\ \text{``P is true or Q is false.''} \qquad$ $ \textbf{(B)}\ \text{``If Q is false then P is true.''} \qquad$ $ \textbf{(C)}\ \text{``If P is false then Q is true.''} \qquad$ $ \textbf{(D)}\ \text{``If Q is true then P is false.''} \qquad$ $ \textbf{(E)}\ \text{``If Q is true then P is true.''}$

1970 Regional Competition For Advanced Students, 3

Tags: 3d geometry , octahedron , ratio , geometry

$E_1$ and $E_2$ are parallel planes and their distance is $p$. (a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$? (b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$ Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way

2019 AMC 8, 18

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The faces on each of two fair dice are numbered 1, 2, 3, 5, 7, and 8. When the two dice are tossed, what is the probability that their sum will be an even number? $\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}$

1990 French Mathematical Olympiad, Problem 5

Tags: triangle , geometry

In a triangle $ABC$, $\Gamma$ denotes the excircle corresponding to $A$, $A',B',C'$ are the points of tangency of $\Gamma$ with $BC,CA,AB$ respectively, and $S(ABC)$ denotes the region of the plane determined by segments $AB',AC'$ and the arc $C'A'B'$ of $\Gamma$. Prove that there is a triangle $ABC$ of a given perimeter $p$ for which the area of $S(ABC)$ is maximal. For this triangle, give an approximate measure of the angle at $A$.

2017 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.

2016 Indonesia TST, 5

Tags: greatest common divisor , combinatorics

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

1964 AMC 12/AHSME, 12

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Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$? $ \textbf{(A)}\ \text{For all x}, x^2 < 0\qquad$ $\textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad$ $\textbf{(C)}\ \text{For no x}, x^2>0\qquad$ ${\textbf{(D)}\ \text{For some x}, x^2>0 }\qquad$ ${{\textbf{(E)}\ \text{For some x}, x^2 \le 0}} $

1992 Czech And Slovak Olympiad IIIA, 6

Tags: altitude , concyclic , circles , geometry

Let $ABC$ be an acute triangle. The altitude from $B$ meets the circle with diameter $AC$ at points $P,Q$, and the altitude from $C$ meets the circle with diameter $AB$ at $M,N$. Prove that the points $M,N,P,Q$ lie on a circle.

2024 Serbia National Math Olympiad, 5

Tags: algebra

Let $n \geq 3$ be a positive integer. Find all positive integers $k$, such that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$f(x)=\cos^k(x)+\cos^k(x+\frac{2\pi}{n})+\ldots +\cos^k(x+\frac{2(n-1)\pi}{n})$$ is constant.

2011 Laurențiu Duican, 2

Tags: linear algebra , nilpotence

Let be a field $ \mathbb{F} $ and two nonzero nilpotent matrices $ M,N\in\mathcal{M}_2\left( \mathbb{F} \right) $ that commute. Show that: [b]a)[/b] $ MN=0 $ [b]b)[/b] there exists a nonzero element $ f\in\mathbb{F} $ such that $ M=fN $ [i]Dorel Miheț[/i]

1994 Czech And Slovak Olympiad IIIA, 2

Tags: cube , polyhedron , volume , max , 3d geometry , projection , geometry

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

2020 Dutch IMO TST, 2

Tags: combinatorics , game , winning strategy , game strategy

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

2016 Saudi Arabia IMO TST, 3

Tags: miscellaneous

Find the number of permutations $ ( a_1, a_2, . \ . \ , a_{2016}) $ of the first $ 2016 $ positive integers satisfying the following two conditions: 1. $ a_{i+1} - a_i \leq 1$ for all $i = 1, 2, . \ . \ . , 2015$, and 2. There are exactly two indices $ i < j $ with $ 1 \leq i < j \leq 2016 $ such that $ a_i = i $ and $ a_j = j$.

2012 Princeton University Math Competition, A1 / B2

Tags: combinatorics

If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.

2019 Bundeswettbewerb Mathematik, 3

Tags: equal angles , incircle , geometry

Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.

2014 Ukraine Team Selection Test, 3

Tags: geometry , complex number , hexagon

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Russian TST 2016, P3

Tags: incircle , geometry

The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. [list=i] [*]Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. [*]Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$. [/list]

2010 Mexico National Olympiad, 2

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.