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How many triangles are in the figure below? [asy]
draw((0,0)--(30,0)--(30,20)--(0,20)--cycle);
draw((15,0)--(15,20));
draw((0,0)--(15,20));
draw((15,0)--(0,20));
draw((15,0)--(30,20));
draw((30,0)--(15,20));
draw((0,10)--(30,10));
draw((7.5,0)--(7.5,20));
draw((22.5,0)--(22.5,20));
[/asy] | 36 | 1/8 |
Consider a 10 x 10 checkerboard made up of 100 squares. How many squares are there that can be formed by the union of one or more squares on the checkerboard? | 385 | 7/8 |
Let $P$ be a point inside regular pentagon $A B C D E$ such that $\angle P A B=48^{\circ}$ and $\angle P D C=42^{\circ}$. Find $\angle B P C$, in degrees. | 84^{\circ} | 1/8 |
A stone is thrown from the Earth's surface at an angle of $60^{\circ}$ to the horizontal with an initial speed of $v_{0}=10 \mathrm{~m} / s$. Determine the radius of curvature of the trajectory at the final point of the flight. The acceleration due to gravity is $g=10 \mathrm{~m} / \mathrm{s}^{2}$. | 20\, | 1/8 |
Determine the number of ways of tiling a $4 \times 9$ rectangle by tiles of size $1 \times 2$. | 6336 | 1/8 |
At a chemistry conference, there are $k$ scientists, consisting of chemists and alchemists, with the number of chemists being more than the number of alchemists. It is known that chemists always tell the truth, while alchemists sometimes tell the truth and sometimes lie. A mathematician who is present at the conference wants to determine for each scientist whether they are a chemist or an alchemist. To do this, the mathematician can ask any scientist the question: "Is such-and-such person a chemist or an alchemist?" (In particular, the mathematician can ask a scientist about themselves.) Prove that the mathematician can determine this with $2 k - 3$ questions. | 2k-3 | 1/8 |
Let \(a, b, c\) be positive integers. All the roots of each of the quadratics
\[ a x^{2}+b x+c, \quad a x^{2}+b x-c, \quad a x^{2}-b x+c, \quad a x^{2}-b x-c \]
are integers. Over all triples \((a, b, c)\), find the triple with the third smallest value of \(a+b+c\). | (1,10,24) | 1/8 |
Find all possible integer solutions for sqrt(x + sqrt(x + ... + sqrt(x))) = y, where there are 1998 square roots. | 0 | 6/8 |
Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB. | \frac{\sqrt{5}}{2} | 6/8 |
Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. | \frac{1}{2} | 5/8 |
Ewan writes out a sequence where he counts by 11s starting at 3. Which number will appear in Ewan's sequence? | 113 | 1/8 |
Square \( M \) has an area of \( 100 \text{ cm}^2 \). The area of square \( N \) is four times the area of square \( M \). The perimeter of square \( N \) is:
(A) \( 160 \text{ cm} \)
(B) \( 400 \text{ cm} \)
(C) \( 80 \text{ cm} \)
(D) \( 40 \text{ cm} \)
(E) \( 200 \text{ cm} \) | 80\, | 1/8 |
The angle at vertex $C$ of a triangle is a right angle. The angle bisector and altitude from $C$ intersect the circumcircle at points $D$ and $E$, respectively. The shorter leg of the triangle is $b$. Prove that the length of the broken line $C D E$ is $b \sqrt{2}$. | b\sqrt{2} | 1/8 |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$. | 3 + \frac{10\sqrt{3}}{3} | 3/8 |
Given the universal set \( U = \{1, 2, 3, 4, 5\} \) and the set \( I = \{X \mid X \subseteq U\} \), two different elements \( A \) and \( B \) are randomly selected from set \(I\). What is the probability that the intersection \( A \cap B \) has exactly 3 elements? | \frac{5}{62} | 3/8 |
A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if
(i) all four sides of the rectangle are segments of drawn line segments, and
(ii) no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000. | 896 | 7/8 |
A number is *interesting*if it is a $6$ -digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers? | 308253 | 1/8 |
Let $m$ be the largest real solution to the equation
\[\dfrac{3}{x-3} + \dfrac{5}{x-5} + \dfrac{17}{x-17} + \dfrac{19}{x-19} = x^2 - 11x - 4\]There are positive integers $a, b,$ and $c$ such that $m = a + \sqrt{b + \sqrt{c}}$. Find $a+b+c$. | 263 | 1/8 |
Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$ , $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$ ). | 3 | 3/8 |
Compute the number of ordered pairs of integers \((x, y)\) such that \(x^{2} + y^{2} < 2019\) and
\[x^{2}+\min(x, y) = y^{2}+\max(x, y).\] | 127 | 7/8 |
If $x>y>0$ , then $\frac{x^y y^x}{y^y x^x}=$ | {\left(\frac{x}{y}\right)}^{y-x} | 1/8 |
What is the smallest prime that is the sum of five different primes? | 43 | 5/8 |
Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\$2436.29$. Determine how much Susie originally invested in Alpha Bank. | 820 | 1/8 |
How many orderings $(a_{1}, \ldots, a_{8})$ of $(1,2, \ldots, 8)$ exist such that $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}-a_{8}=0$ ? | 4608 | 7/8 |
Given vectors
$$
\boldsymbol{m}=\left(\cos \frac{x}{2},-1\right), \boldsymbol{n}=\left(\sqrt{3} \sin \frac{x}{2}, \cos ^{2} \frac{x}{2}\right) ,
$$
define the function $f(x)=\boldsymbol{m} \cdot \boldsymbol{n}+1$.
1. If $x \in\left[0, \frac{\pi}{2}\right]$ and $f(x)=\frac{11}{10}$, find the value of $\cos x$.
2. In $\triangle ABC$, let the sides opposite to $\angle A$, $\angle B$, and $\angle C$ be $a$, $b$, and $c$, respectively. If $2 b \cos A \leqslant 2 c-\sqrt{3} a$, find the range of values for $f(B)$. | (0,\frac{1}{2}] | 7/8 |
If in applying the [quadratic formula](https://artofproblemsolving.com/wiki/index.php/Quadratic_formula) to a [quadratic equation](https://artofproblemsolving.com/wiki/index.php/Quadratic_equation)
\[f(x) \equiv ax^2 + bx + c = 0,\]
it happens that $c = \frac{b^2}{4a}$, then the graph of $y = f(x)$ will certainly:
$\mathrm{(A) \ have\ a\ maximum } \qquad \mathrm{(B) \ have\ a\ minimum} \qquad$ $\mathrm{(C) \ be\ tangent\ to\ the\ x-axis} \qquad$ $\mathrm{(D) \ be\ tangent\ to\ the\ y-axis} \qquad$ $\mathrm{(E) \ lie\ in\ one\ quadrant\ only}$ | \mathrm{(C)\be\tangent\to\the\x-axis} | 1/8 |
Let \\(\alpha\\) be an acute angle. If \\(\sin \left(\alpha+ \frac {\pi}{6}\right)= \frac {3}{5}\\), then \\(\cos \left(2\alpha- \frac {\pi}{6}\right)=\\) ______. | \frac {24}{25} | 7/8 |
Rectangle $HOMF$ has $HO=11$ and $OM=5$ . Triangle $ABC$ has orthocenter $H$ and circumcenter $O$ . $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$ . Find the length of $BC$ . | 28 | 5/8 |
If line $l_1: ax+2y+6=0$ is parallel to line $l_2: x+(a-1)y+(a^2-1)=0$, then the real number $a=$ . | -1 | 4/8 |
You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$ . Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$ ?
For example, if $a = [-1, 2, 2]$ , then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$ .
*(Proposed by Anton Trygub)* | 2^{n-1} | 3/8 |
Kolya found a fun activity: he rearranges the digits of the number 2015, after which he puts a multiplication sign between any two digits and calculates the value of the resulting expression. For example: \(150 \cdot 2 = 300\), or \(10 \cdot 25 = 250\). What is the largest number he can get as a result of such a calculation? | 1050 | 4/8 |
Let \(a, b\) be nonzero complex numbers, and \(\frac{a}{b}\) is not a real number. Define:
\[
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\},
\]
\[
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]
Find the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) change. | 6 | 3/8 |
Team A has a probability of $$\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match. | \frac{19}{27} | 4/8 |
In an olympiad, 2006 students participated. It was found that a student, Vasia, solved only one out of the six problems. Additionally, the number of participants who solved at least 1 problem is 4 times greater than those who solved at least 2 problems;
the number who solved at least 2 problems is 4 times greater than those who solved at least 3 problems;
the number who solved at least 3 problems is 4 times greater than those who solved at least 4 problems;
the number who solved at least 4 problems is 4 times greater than those who solved at least 5 problems;
the number who solved at least 5 problems is 4 times greater than those who solved all 6.
How many students did not solve any problem? | 982 | 7/8 |
Professor Rackbrain recently asked his young friends to find all five-digit perfect squares for which the sum of the numbers formed by the first two digits and the last two digits equals a perfect cube. For example, if we take the square of the number 141, which is equal to 19881, and add 81 to 19, we get 100 – a number that unfortunately is not a perfect cube.
How many such solutions exist? | 3 | 1/8 |
Given a regular pentagon with side length 1, draw the two diagonals from one vertex. This creates a net for a tetrahedron with a regular triangular base. What is the volume of this tetrahedron? | \frac{1+\sqrt{5}}{24} | 1/8 |
The Matini company released a special album with the flags of the $ 12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is $50\%$ . | 3 | 4/8 |
Find the largest positive real $ k$ , such that for any positive reals $ a,b,c,d$ , there is always:
\[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\] | 174960 | 1/8 |
The digits of the four-digit decimal number \(\overline{a b c d}\) satisfy \(a > b > c > d\). The digits can be rearranged in some order to form the difference \(\overline{a b c d} - \overline{d c b a}\). What is this four-digit number? | 7641 | 1/8 |
Class 5-1 has 40 students. In a math exam, the average score of the top 8 students is 3 points higher than the average score of the entire class. The average score of the other students is $\qquad$ points lower than the average score of the top 8 students. | 3.75 | 6/8 |
If the equation \( x^{3} - 3x^{2} - 9x = a \) has exactly two different real roots in the interval \([-2, 3]\), then the range of the real number \( a \) is \(\quad\) . | [-2,5) | 3/8 |
Petya conceived a natural number and wrote the sums of each pair of its digits on the board. Then he erased some sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have conceived? | 2000 | 1/8 |
Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ . | 1/2 | 1/8 |
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?
(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).) | 10 | 2/8 |
In an $8 \times 8$ grid filled with different natural numbers, where each cell contains only one number, if a cell's number is greater than the numbers in at least 6 other cells in its row and greater than the numbers in at least 6 other cells in its column, then this cell is called a "good cell". What is the maximum number of "good cells"? | 16 | 4/8 |
The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\). | \frac{1}{2} + \sqrt{2} | 6/8 |
If altitude $CD$ is $\sqrt3$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?
[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(3)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$30^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,NE);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(210)--D + sqrt(2)*.1*dir(165)--D+.1*dir(120));
[/asy] | 2\sqrt{3} | 5/8 |
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$? | 36.8 | 5/8 |
When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is
$\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$ | (D)\826 | 1/8 |
Given the expression \[2 - (-3) - 4 - (-5) - 6 - (-7) \times 2,\] calculate its value. | -14 | 1/8 |
Identify the maximum value of the parameter \( a \) for which the system of equations
\[
\left\{\begin{array}{l}
y = 1 - \sqrt{x} \\
a - 2(a - y)^2 = \sqrt{x}
\end{array}\right.
\]
has a unique solution. | 2 | 7/8 |
For a real number $x$, let $[x]$ be $x$ rounded to the nearest integer and $\langle x\rangle$ be $x$ rounded to the nearest tenth. Real numbers $a$ and $b$ satisfy $\langle a\rangle+[b]=98.6$ and $[a]+\langle b\rangle=99.3$. Compute the minimum possible value of $[10(a+b)]$. | 988 | 1/8 |
The difference between the longest and shortest diagonals of a regular n-gon equals its side. Find all possible values of n. | 9 | 2/8 |
There is one electrical outlet connected to the network, two power strips with three outlets each, and one desk lamp included. Neznaika randomly plugged all three plugs into 3 out of the 7 outlets. What is the probability that the lamp will light up? | \frac{13}{35} | 1/8 |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi}{4}$, find the area of $\Delta ABC$. | \sqrt{3} +1 | 1/8 |
Given the function $f(x)=\ln (1+x)- \frac {x(1+λx)}{1+x}$, if $f(x)\leqslant 0$ when $x\geqslant 0$, calculate the minimum value of $λ$. | \frac {1}{2} | 3/8 |
There are 300 balls in total, consisting of white balls and red balls, and 100 boxes. Each box contains 3 balls. There are 27 boxes that contain 1 white ball. There are 42 boxes that contain 2 or 3 red balls. The number of boxes that contain 3 white balls is equal to the number of boxes that contain 3 red balls. How many white balls are there in total? | 158 | 3/8 |
The ring road is divided by kilometer posts, and it is known that the number of posts is even. One of the posts is painted yellow, another is painted blue, and the rest are painted white. The distance between posts is defined as the length of the shortest arc connecting them. Find the distance from the blue post to the yellow post if the sum of the distances from the blue post to the white posts is 2008 km. | 17 | 6/8 |
A grid of size \( n \times n \) is composed of small squares with side length 1. Each small square is either colored white or black. The coloring must satisfy the condition that in any rectangle formed by these small squares, the four corner squares are not all the same color. What is the largest possible value of the positive integer \( n \)?
A. 3
B. 4
C. 5
D. 6 | 4 | 1/8 |
For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$ . Find all $n$ for which the sequence $a_k$ increases starting from some number. | 2 | 3/8 |
Given \( a, b, c \in \mathbf{R}_{+} \) and \( a^{2} + b^{2} + c^{2} = 1 \), let
\[ M = \max \left\{ a + \frac{1}{b}, b + \frac{1}{c}, c + \frac{1}{a} \right\}, \]
find the minimum value of \( M \). | \frac{4\sqrt{3}}{3} | 1/8 |
Let \(\mathcal{C}_1\) and \(\mathcal{C}_2\) be two concentric circles, with \(\mathcal{C}_2\) inside \(\mathcal{C}_1\). Let \(A\) be a point on \(\mathcal{C}_1\) and \(B\) be a point on \(\mathcal{C}_2\) such that the line segment \(AB\) is tangent to \(\mathcal{C}_2\). Let \(C\) be the second point of intersection of the line segment \(AB\) with \(\mathcal{C}_1\), and let \(D\) be the midpoint of \(AB\). A line passing through \(A\) intersects \(\mathcal{C}_2\) at \(E\) and \(F\) such that the perpendicular bisectors of segments \(DE\) and \(CF\) intersect at a point \(M\) on the line segment \(AB\). Determine the ratio \(AM / MC\). | 1 | 1/8 |
The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum? | 14 | 6/8 |
Determine all primes \( p \) such that
\[ 5^p + 4 \cdot p^4 \]
is a perfect square, i.e., the square of an integer. | 5 | 5/8 |
Let \( a_{n} \) be a quasi-injective sequence of integers (i.e., there exists \( C > 0 \) such that the equation \( a_{n} = k \) has fewer than \( C \) solutions for any integer \( k \)). Show that if \( a_{n} \) has only a finite number of prime divisors, then
$$
\sum_{n} \frac{1}{a_{n}}<\infty
$$ | \sum_{n}\frac{1}{a_{n}}<\infty | 7/8 |
Let \( a, b, c, d, e, f \) be integers selected from the set \( \{1,2, \ldots, 100\} \), uniformly and at random with replacement. Set
\[ M = a + 2b + 4c + 8d + 16e + 32f. \]
What is the expected value of the remainder when \( M \) is divided by 64? | 31.5 | 1/8 |
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$ . Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$ .
How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime? | 2 | 1/8 |
In the rhombus \(ABCD\), the angle \(BCD\) is \(135^{\circ}\), and the sides are 8. A circle touches the line \(CD\) and intersects side \(AB\) at two points located 1 unit away from \(A\) and \(B\). Find the radius of this circle. | \frac{41 \sqrt{2}}{16} | 5/8 |
Let \( d \) be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than \( d \) degrees. What is the minimum possible value for \( d \) ? | 120 | 6/8 |
Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements. | 18 | 7/8 |
The base of a prism is a trapezoid. Express the volume of the prism in terms of the areas \( S_{1} \) and \( S_{2} \) of the parallel lateral faces and the distance \( h \) between them. | \frac{(S_1+S_2)}{2} | 3/8 |
Let \( f(x) = x^2 + a \). Define \( f^1(x) = f(x) \), \( f^n(x) = f\left(f^{n-1}(x)\right) \) for \( n = 2, 3, \ldots \). Let \( M = \left\{a \in \mathbf{R} \mid \) for all positive integers \( n, \left| f^n(0) \right| \leq 2 \right\} \). Prove that \( M = \left[-2, \frac{1}{4}\right] \). | [-2,\frac{1}{4}] | 2/8 |
At a nursery, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies? | \frac{1003}{2} | 3/8 |
Given $f(x)= \frac{1}{2^{x}+ \sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$. | 3 \sqrt {2} | 2/8 |
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$?
$\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$ | \textbf{(B)}20 | 1/8 |
In the diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$ if the length of segment $CD$ is 39 units?
[asy]
draw((0,0)--(39,0)--(10,18)--(0,0)); // Adjusted for new problem length
dot((0,0));
label("$A$",(0,0),SW);
label("9",(4.5,0),S); // New base length of ABC
dot((9,0));
label("$C$",(9,0),S);
label("39",(24,0),S); // New length for CD
dot((39,0));
label("$D$",(39,0),SE);
dot((10,18)); // Adjusted location of B to maintain proportionality
label("$B$",(10,18),N);
draw((9,0)--(10,18));
[/asy] | 156 | 6/8 |
Determine all positive integers \( n \) for which there exists a set \( S \) with the following properties:
(i) \( S \) consists of \( n \) positive integers, all smaller than \( 2^{n-1} \);
(ii) for any two distinct subsets \( A \) and \( B \) of \( S \), the sum of the elements of \( A \) is different from the sum of the elements of \( B \). | n\ge4 | 1/8 |
Find all pairs of prime numbers \((p, q)\) such that
$$
\left(3 p^{q-1}+1\right) \mid \left(11^{p}+17^{p}\right).
$$ | (3,3) | 1/8 |
Determine the largest positive integer $n$ such that the following statement is true:
There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does. | 3 | 1/8 |
Let the complex number \( z \) satisfy the condition \( |z - i| = 1 \), with \( z \neq 0 \) and \( z \neq 2i \). Given that the complex number \( \omega \) is such that \( \frac{\omega - 2i}{\omega} \cdot \frac{z}{z - 2i} \) is a real number, determine the principal value range of the argument of the complex number \( \omega - 2 \). | [\pi-\arctan\frac{4}{3},\pi] | 1/8 |
How many diagonals in a regular 32-sided polygon are not parallel to any of its sides? | 240 | 1/8 |
Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy:
(1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards.
(2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards.
Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$ .
*Remark*: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$ .
*Proposed by usjl* | N^2 | 3/8 |
How many lattice points lie on the hyperbola $x^2 - y^2 = 3^4 \cdot 17^2$? | 30 | 4/8 |
Consider all prisms whose base is a convex 2015-gon.
What is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices? | 2017 | 4/8 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1014 | 3/8 |
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
*Dumitru Bușneag* | P(x)=ax+b | 7/8 |
For \( i=1,2, \cdots, n \), given \( \left|x_{i}\right|<1 \), and \( \left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=2005+\left| x_{1} + x_{2}+\cdots+x_{n} \right| \). Find the smallest positive integer \( n \). | 2006 | 7/8 |
How many solutions does the equation \((a-1)(\sin 2 x + \cos x) + (a+1)(\sin x - \cos 2 x) = 0\) (where the parameter \(a < 0\)) have in the interval \((-π, π)\) ? | 4 | 1/8 |
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ - $y$ plane.
Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ *2010 Tohoku University entrance exam/Economics, 2nd exam* | \frac{8}{3} | 7/8 |
Convert the base 2 number \(1011111010_2\) to its base 4 representation. | 23322_4 | 4/8 |
On each square of a chessboard, one can choose to place 0 or 1 tokens, such that the number of tokens in each row, each column, and each diagonal is at most 4. How many tokens can be placed at most? | 32 | 7/8 |
Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$ If $g(81) = 81,$ compute $[g(9)]^4.$ | 81 | 6/8 |
For a real number \( x \), \([x]\) denotes the greatest integer less than or equal to \( x \). Given a sequence of positive numbers \( \{a_n\} \) such that \( a_1 = 1 \) and \( S_n = \frac{1}{2} \left( a_n + \frac{1}{a_n} \right) \), where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \), then \(\left[ \frac{1}{S_1} + \frac{1}{S_2} + \cdots + \frac{1}{S_{100}} \right] = \, \). | 18 | 5/8 |
Adam and Sarah start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Sarah travels west at 5 mph. After how many hours are they 85 miles apart? | 7.6 | 2/8 |
Let \( A B C D \) be a convex quadrilateral with \( AB=5, BC=6, CD=7\), and \( DA=8 \). Let \( M, P, N, Q \) be the midpoints of sides \( AB, BC, CD, \) and \( DA \) respectively. Compute \( MN^{2} - PQ^{2} \). | 13 | 7/8 |
Find all triples of positive integers (x, y, z) such that \( x \leq y \leq z \), and
\[ x^{3}\left(y^{3} + z^{3}\right) = 2012(x y z + 2). \] | (2,251,252) | 1/8 |
Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. | \frac{250}{7} | 1/8 |
Points \( M \) and \( N \) divide side \( AC \) of triangle \( ABC \) into three equal parts, each of which is 5, with \( AB \perp BM \) and \( BC \perp BN \). Find the area of triangle \( ABC \). | \frac{75 \sqrt{3}}{4} | 1/8 |
Let \( \triangle ABC \) and \( \triangle PQR \) be two triangles. If \( \cos A = \sin P \), \( \cos B = \sin Q \), and \( \cos C = \sin R \), what is the largest angle (in degrees) among the six interior angles of the two triangles? | 135 | 6/8 |
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$
\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]
Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$ | -1989 | 5/8 |
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