Olympiad problems

2003 CHKMO, 4

Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

2003 All-Russian Olympiad, 3

Tags: combinatorics unsolved , combinatorics

There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.

1952 AMC 12/AHSME, 9

Tags:

If $ m \equal{} \frac {cab}{a \minus{} b}$, then $ b$ equals: $ \textbf{(A)}\ \frac {m(a \minus{} b)}{ca} \qquad\textbf{(B)}\ \frac {cab \minus{} ma}{ \minus{} m} \qquad\textbf{(C)}\ \frac {1}{1 \plus{} c} \qquad\textbf{(D)}\ \frac {ma}{m \plus{} ca}$ $ \textbf{(E)}\ \frac {m \plus{} ca}{ma}$

2009 F = Ma, 25

Tags: 2009 , problem 25

Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii $r_\text{1}$ and $r_\text{2}$. The discs are given angular velocities of magnitudes $\omega_\text{1}$ and $\omega_\text{2}$, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods. [asy] //Code by riben, Improved by CalTech_2023 // Solids import solids; //bigger cylinder draw(shift(0,0,-1)*scale(0.1,0.1,0.59)*unitcylinder,surfacepen=white,black); draw(shift(0,0,-0.1)*unitdisk, surfacepen=black); draw(unitdisk, surfacepen=white,black); draw(scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); //smaller cylinder draw(rotate(5,X)*shift(-2,3.2,-1)*scale(0.1,0.1,0.6)*unitcylinder,surfacepen=white,black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.55)*unitdisk, surfacepen=black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.6)*unitdisk, surfacepen=white,black); draw(rotate(5,X)*shift(-2,3.2,-0.2)*scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); // Lines draw((0,-2)--(1,-2),Arrows(size=5)); draw((4,-2)--(4.7,-2),Arrows(size=5)); // Labels label("r1",(0.5,-2),S); label("r2",(4.35,-2),S); // Curved Lines path A=(-0.694, 0.897)-- (-0.711, 0.890)-- (-0.742, 0.886)-- (-0.764, 0.882)-- (-0.790, 0.873)-- (-0.815, 0.869)-- (-0.849, 0.867)-- (-0.852, 0.851)-- (-0.884, 0.844)-- (-0.895, 0.837)-- (-0.904, 0.824)-- (-0.879, 0.800)-- (-0.841, 0.784)-- (-0.805, 0.772)-- (-0.762, 0.762)-- (-0.720, 0.747)-- (-0.671, 0.737)-- (-0.626, 0.728)-- (-0.591, 0.720)-- (-0.556, 0.715)-- (-0.504, 0.705)-- (-0.464, 0.700)-- (-0.433, 0.688)-- (-0.407, 0.683)-- (-0.371, 0.685)-- (-0.316, 0.673)-- (-0.271, 0.672)-- (-0.234, 0.667)-- (-0.192, 0.664)-- (-0.156, 0.663)-- (-0.114, 0.663)-- (-0.070, 0.660)-- (-0.033, 0.662)-- (0.000, 0.663)-- (0.036, 0.663)-- (0.067, 0.665)-- (0.095, 0.667)-- (0.125, 0.666)-- (0.150, 0.673)-- (0.187, 0.675)-- (0.223, 0.676)-- (0.245, 0.681)-- (0.274, 0.687)-- (0.300, 0.696)-- (0.327, 0.707)-- (0.357, 0.709)-- (0.381, 0.718)-- (0.408, 0.731)-- (0.443, 0.740)-- (0.455, 0.754)-- (0.458, 0.765)-- (0.453, 0.781)-- (0.438, 0.795)-- (0.411, 0.809)-- (0.383, 0.817)-- (0.344, 0.829)-- (0.292, 0.839)-- (0.254, 0.846)-- (0.216, 0.851)-- (0.182, 0.857)-- (0.153, 0.862)-- (0.124, 0.867); draw(shift(0.2,0)*A,EndArrow(size=5)); path B=(2.804, 0.844)-- (2.790, 0.838)-- (2.775, 0.838)-- (2.758, 0.831)-- (2.740, 0.831)-- (2.709, 0.827)-- (2.688, 0.825)-- (2.680, 0.818)-- (2.660, 0.810)-- (2.639, 0.810)-- (2.628, 0.803)-- (2.618, 0.799)-- (2.604, 0.790)-- (2.598, 0.778)-- (2.596, 0.769)-- (2.606, 0.757)-- (2.630, 0.748)-- (2.666, 0.733)-- (2.696, 0.721)-- (2.744, 0.707)-- (2.773, 0.702)-- (2.808, 0.697)-- (2.841, 0.683)-- (2.867, 0.680)-- (2.912, 0.668)-- (2.945, 0.665)-- (2.973, 0.655)-- (3.010, 0.648)-- (3.040, 0.647)-- (3.069, 0.642)-- (3.102, 0.640)-- (3.136, 0.632)-- (3.168, 0.629)-- (3.189, 0.627)-- (3.232, 0.619)-- (3.254, 0.624)-- (3.281, 0.621)-- (3.328, 0.618)-- (3.355, 0.618)-- (3.397, 0.617)-- (3.442, 0.616)-- (3.468, 0.611)-- (3.528, 0.611)-- (3.575, 0.617)-- (3.611, 0.619)-- (3.634, 0.625)-- (3.666, 0.622)-- (3.706, 0.626)-- (3.742, 0.635)-- (3.772, 0.635)-- (3.794, 0.641)-- (3.813, 0.646)-- (3.837, 0.654)-- (3.868, 0.659)-- (3.886, 0.672)-- (3.903, 0.681)-- (3.917, 0.688)-- (3.931, 0.697)-- (3.943, 0.711)-- (3.951, 0.720)-- (3.948, 0.731)-- (3.924, 0.745)-- (3.900, 0.757)-- (3.874, 0.774)-- (3.851, 0.779)-- (3.821, 0.779)-- (3.786, 0.786)-- (3.754, 0.792)-- (3.726, 0.797)-- (3.677, 0.806)-- (3.642, 0.812); draw(shift(0.7,0)*B,EndArrow(size=5)); [/asy] (A) $\omega_\text{1}^2r_\text{1}=\omega_\text{2}^2r_\text{2}$ (B) $\omega_\text{1}r_\text{1}=\omega_\text{2}r_\text{2}$ (C) $\omega_\text{1}r_\text{1}^2=\omega_\text{2}r_\text{2}^2$ (D) $\omega_\text{1}r_\text{1}^3=\omega_\text{2}r_\text{2}^3$ (E) $\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4$

2019 Iran Team Selection Test, 6

Tags: number theory

$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$ $$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$ prove that $a_{n}=b_{n}$ for $n\geq 0$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.) [i]Proposed by Yahya Motevassel[/i]

2011 Postal Coaching, 2

Tags: modular arithmetic , number theory unsolved , number theory

Let $S(k)$ denote the digit-sum of a positive integer $k$(in base $10$). Determine the smallest positive integer $n$ such that \[S(n^2 ) = S(n) - 7\]

2019 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorics

Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define [i]restructing[/i] to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings. [i] (Т. Зубов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

MOAA Team Rounds, 2021.5

Tags: MOAA 2021 , team

Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral? [asy] size(4cm); fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3)); draw((0,0)--(0,12)--(-5,0)--cycle); draw((0,0)--(8,0)--(0,6)); label("5", (-2.5,0), S); label("13", (-2.5,6), dir(140)); label("6", (0,3), E); label("8", (4,0), S); [/asy] [i]Proposed by Nathan Xiong[/i]

2010 IFYM, Sozopol, 2

Tags: geometry

Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

2022 Canadian Junior Mathematical Olympiad, 4

Tags: number theory

I think we are allowed to discuss since its after 24 hours How do you do this Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function

2014 Tuymaada Olympiad, 2

Tags: geometry , geometric transformation , homothety , reflection , geometry unsolved , Tuymaada

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

2019 CHMMC (Fall), 7

Tags: number theory

Let $S$ be the set of all positive integers $n$ satisfying the following two conditions: $\bullet$ $n$ is relatively prime to all positive integers less than or equal to $\frac{n}{6}$ $\bullet$ $2^n \equiv 4$ mod $n$ What is the sum of all numbers in $S$?

2016 Indonesia TST, 1

Tags: number theory , Sets , abstract algebra

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

1992 IMO Longlists, 53

Tags: number theory , IMO

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

2018 Belarus Team Selection Test, 1.2

Tags: geometry , parallelogram , ratio

Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively. Find all possible values of the ratio $d_1:d_2$. [i](I. Voronovich)[/i]

LMT Speed Rounds, 12

Tags: combinatorics

Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn. [i]Proposed by Samuel Tsui[/i]

1959 AMC 12/AHSME, 36

Tags: trigonometry , AMC , Triangle Geometry , geometry , AMC 12

The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. The shortest side is: $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12 $

2011 National Olympiad First Round, 16

Tags:

There are $2011$ stones, whose weights are positive integers. If it is possible to divide these stones into $n$ groups not containing two stones with one weighs two times of the other, what is the least possible value of $n$? $\textbf{(A)}\ 102 \qquad\textbf{(B)}\ 51 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None}$

1991 AMC 12/AHSME, 23

Tags: geometry , ratio , AMC

If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0); draw(A--E--B--F--C--D--A--F^^E--D--B); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, W); label("F", F, S); label("H", (.8,0.6)); label("I", (0.4,1.4)); [/asy] $ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $

2015 Regional Competition For Advanced Students, 4

Tags: Austria , geometry

Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$. (a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular. (Walther Janous)

2000 Miklós Schweitzer, 6

Tags: college contests , Miklos Schweitzer , Borel set , real analysis

Suppose the real line is decomposed into two uncountable Borel sets. Prove that a suitable translated copy of the first set intersects the second in an uncountable set.

2019 China Team Selection Test, 4

Tags: number theory

Prove that there exist a subset $A$ of $\{1,2,\cdots,2^n\}$ with $n$ elements, such that for any two different non-empty subset of $A$, the sum of elements of one subset doesn't divide another's.

2023 IFYM, Sozopol, 6

Tags:

Let $S$ be a set of real numbers. We say that $S$ is [i]strong[/i] if for any two distinct $a$ and $b$ from $S$, the number $a^2 + b\sqrt{2023}$ is rational. We say that $S$ is [i]very strong[/i] if for every $a$ from $S$, the number $a\sqrt{2023}$ is rational. a) Prove that if $S$ is a very strong set, then it is also strong. b) Find the smallest natural number $k$ such that every strong set of $k$ distinct real numbers is very strong.

Kvant 2022, M2722

Tags: geometry , Tournament of Towns

Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

2014 District Olympiad, 4

Tags: floor function , algebra proposed , algebra

Let $n\geq2$ be a positive integer. Determine all possible values of the sum \[ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor \] where $x_i\in \mathbb{R}$ satisfying $\lfloor{x_i}\rfloor=i$ for $i=1,2,\ldots n$.